INtegral

CALCULUS 2:INTEGRAL CALCULUST h i s m a t e r i a l i s e x c l u s i v e l y f o r t h e u s e o f F e l l o w s h i p B a p t i s t C o l l e g e a n d m u s t n o t b e p u b l i s h e d , d i s t r i b u t e d , o r t r a n s m i t t e d o u t s i d e t h e i n s t i t u t i o n . R e p r o d u c t i o n , s t o r i n g i n a r e t r i e v a l s y s t e m , o r a n y o t h e r f o r m o f e l e c t r o n i c o r m e c h a n i c a l d i s s e m i n a t i o n i s s t r i c t l y p r o h i b i t e d w i t h o u t p r i o r w r i t t e n a u t h o r i z a t i o n f r o m F B C .TEACHER’S INFORMATIONENGMATH 122ENGMATH 122 F E L L O W S H I P B A P T I S T C O L L E G E C O L L E G E O F E N G I N E E R I N G A N D C O M P U T E R S T U D I E S A . Y . 2 0 2 6 | M I D Y E A R ENGR. CELPHY L. TRIMUCHA CONTACT INFORMATION Bachelor of Science in Electronics Engineering Consultation Schedule Friday (04:00 - 05:30 PM)trimucha_celphy@fbc.ph.education+639 61 812 7988Celphy L. TrimuchaM| 01:00 AM-07:00 PM | SB 102 TWTH | 01:00 PM-07:00 PM | PR2B This module belongs to:Reach your dreams andMake a Difference.

It stretches like a ribbon across the grid, threading through spaces infinitely small, bending and weaving, framing the empty void, until it finds its bounds to hold. Invisible limits carry it onward, guiding it past each partition, shaping it with patience, giving it area and accumulation without ever revealing the infinite parts. Rectangles rise like sentinels in the distance, reaching upward to touch the slope, welcoming the wandering variables, turning fragments into presence, and chaos into a collective whole. Every delta leaves a trace on the axis, every interval holds a secret, and every summation builds a bridge from change to completion. The line may twist and falter, the approximation may stumble, yet in the quiet persistence of its gathering, it discovers form and finds resolution. In the delicate space between piece and constant, accumulation becomes a function, and mathematics becomes poetry. For it is not the sliver that matters, nor the rate at which it changes, but the completeness it awakens, the truth that it inspires, and the bridge it builds between the fragment and the whole."Between Fragments and the Whole" Before a curve becomes a total, it flows quietly, seeking a sum.

Dear Future Engineer, Like designing a structure, writing a program, or tuning a communication network—it begins with a strong foundation and grows through deliberate, precise work. At first glance, the symbols and equations of calculus may seem abstract, but within them lies the language that shapes bridges, powers software, optimizes networks, and unlocks the secrets of motion, change, and propagation. This mathematical journey unites you across disciplines in your pursuit of clarity, precision, and purpose: For Civil Engineers (CE): You will discover how calculus helps you understand the continuous forces beneath massive structures, analyze the distribution of stress, calculate the fluid volumes within changing landscapes, and map the precise curves of the physical world. For Computer Engineers (CpE): It sharpens your algorithmic logic, models complex digital and continuous systems, and helps you optimize data processing, signal transformations, and machine learning solutions in an ever-evolving digital space. For Electronics and Communication Engineers (ECE): It reveals how varying currents accumulate into charge, how continuous waves carry information through space, and how complex waveforms are broken down and reconstructed to bridge distances and connect minds. You will learn that integrals are more than numbers—they represent accumulation, transformation, and the beauty of the whole formed by the sum of its infinitely small parts. In many ways, they mirror your growth: the small daily lessons, late-night efforts, and sudden breakthroughs that gradually add up to deep engineering mastery over time. Yes, there will be moments of difficulty—when concepts feel heavy, equations refuse to balance, or your progress seems slow. But just like engineers do not abandon a project when faced with structural, digital, or signal complexity, you too must persist, refine, and rise. Calculus rewards the patient and the curious, those who take the time to see the bigger picture and trust the process of continuous summation. Let this course remind you that every curve has an area, every system a steady state, every problem a solution, and every effort a purpose. Grounded in faith, let your studies be an offering—of discipline, excellence, and gratitude for the distinct talents you’ve been given. Let your work in calculus prepare you not just for board exams, but for real-world impact: building, designing, coding, communicating, and creating for the good of others and the glory of God. Keep integrating your knowledge. Keep calculating your growth. And never forget: every great engineer once sat where you are— pencil in hand, doubts in mind, but limitless potential in heart. Wishing you clarity, confidence, and courage on this mathematical journey!SINCERELY, ENGR. CELPHY L. TRIMUCHA, ECT AUTHOR AND EDUCATOR COLLEGE OF ENGINEERING AND COMPUTER STUDIES

INTEGRAL CALCULUS DEPARTMENT College of Engineering and Computer Studies PROGRAM Bachelor of Science in Electronics Engineering COURSE CODE EngMath 122 DESCRIPTIVE TITLE CALCULUS 2 CREDIT UNITS 4 units PREREQUISITE/S (If applicable) Calculus 1 CLASS SCHEDULE (Time/Day/Room) M 01:00 PM-07:00 PM SB 102 TWTh 01:00 PM-07:00 PM PR 2B CONSULTATION SCHEDULE F 04:00 – 5:00 PM COURSE DESCRIPTION The course will focus on topics such as area under curves, definite and indefinite integrals, applications of integration in geometry, area between curves, volumes of solids of revolution (using Disk, Washer, and Shell methods), arc length, and surface area of revolved surfaces. Emphasis will also be placed on integration techniques and their use in solving real-world engineering problems. Moreover, this subject will help students develop a reflective and discerning approach to the mathematical foundations of engineering. This includes exploring how small contributions combine to produce meaningful results, reflecting on the role of precision and structure in problem-solving, and discerning how mathematical thinking can be used responsibly and ethically in designing systems that serve the greater good. COURSE GUIDEGENERAL OBJECTIVES The course EngMath 122 prepares you, as a future engineer, to demonstrate all the intended learning outcomes of this three-unit course in accordance with the Commission on Higher Education (CHED) Memorandum Orders (CMO) for your respective degree program. This course covers the fundamental concepts of definite and indefinite integrals and their applications to area, volume, arc length, surface area, and problem-solving in engineering contexts. At the end of the course, you should be able to: 🔹 Conceptualize and apply definite integrals to solve problems involving area, volume, arc length, and surface area. 🔹 Use appropriate integration techniques such as the Disk, Washer, and Shell methods to model and analyze engineering problems. 🔹 Evaluate real-world applications of integrals in the context of civil and computer engineering systems, including construction, design, signal processing, and data analysis. 🔹 Analyze and reflect on how small elements, when integrated, contribute to larger systems — both mathematically and ethically — and make informed, values-based decisions when applying mathematical tools in professional practice.

Computation for Final Grade Basis of Term Grades Term Percentage Components Percentage Prelim 20% Major Examination 40% Midterm 25% Class Performance 60% Pre-final 25% - Quizzes (25%) Final 30% - Projects/Requirements (20%) Final Grade Sum of Percentages of Grades per Term - Oral Recitation (10%) - ORLE (5%) Total 100%CLASSROOM RULES COLLEGE OF ENGINEERING AND COMPUTER STUDIESGRADING SYSTEM

In mathematics, Integral Calculus teaches us how to measure the accumulated effect of small, continuous changes—such as computing the exact area under a curve, the volume of a solid, or the total work done by a force. It allows engineers to analyze how individual elements contribute to a complete system. Whether it's modeling the amount of concrete needed for a structure or analyzing energy flow in a circuit, integration is essential for creating designs that are both functional and efficient. In our spiritual lives, we sometimes struggle to make sense of scattered events—unexpected setbacks, quiet successes, and moments that feel too small to matter. But just as integration adds up tiny parts to reveal a complete picture, God weaves every moment into His greater purpose. As Romans 8:28 reminds us, “And we know that in all things God works for the good of those who love Him, who have been called according to His purpose.” Just like an engineer learns to trust the process of building something piece by piece, we are called to trust that every season of our lives—every triumph and trial—has value in God’s perfect plan. When we align our decisions, attitudes, and actions with His will, we begin to see how even the smallest moments shape a meaningful, God-honoring life. As you study integral calculus, may you not only gain analytical skills but also a deeper appreciation for how both math and faith teach us that wholeness comes from unity, patience, and purpose. Let this journey remind you that the One who created the universe also integrates every part of your life with divine precision and care. 01 INTRODUCTION TO INTEGRAL CALCULUSLEARNING OBJECTIVESBy the end of this module, students should be able to:Explain the concept of an antiderivative as the reverse of a derivative. Define and interpret an indefinite integral as a general form of all possible antiderivatives. Find basic antiderivatives using the power rule and sum/difference rules. Reflect on how finding antiderivatives shows the importance of tracing back to origins and understanding how things accumulate over time. COLLEGE OF ENGINEERING AND COMPUTER STUDIES

LESSON OVERVIEW: Understanding antiderivatives is essential in integral calculus, as it forms the basis for solving problems involving accumulation and reconstruction of original functions. By learning how to reverse the process of differentiation, students gain the ability to find a function when its rate of change is known— an important concept in physics, engineering, and other applied fields. This foundational skill allows for precise solutions in areas like motion analysis, populationmodeling,andenergy calculations. Mastering antiderivatives also plays a crucial role in advancing mathematical and engineering applications. It supports the development of more complex integration techniques, aids in solving differential equations, and underpins many numerical methods used in simulations and design. Whether in civil engineering, computer engineering, or applied research, the ability to find and interpret antiderivatives ensures accurate modeling of real-world phenomena and effective problem-solving. As students progress in calculus, a strong grasp of antiderivatives remains fundamental for deeper exploration of definite integrals andpractical applications. FELLOWSHIP BAPTIST COLLEGE 02 Building Understanding, Step by Step VALUES INTEGRATIONMOTIVATIONAL ACTIVITY Objective: To stimulate curiosity and appreciation for communication parameters and antenna design by exploring real-world signal transmission challenges. Instructions: Imagine you're walking on a sandy beach. As you walk, you leave footprints. These footprints represent the "rate of change" of your position – how your presence affects the sand at each step. Now, imagine someone else comes along after you, only seeing the footprints. Their task is to figure out your exact path, or your original position, from just these impressions. They might know you were heading north, but they won't know where you started or if you took a break at some point and then resumed from a slightly different spot, unless they have more information. Question for Reflection: If the footprints represent the "derivative" of your path, what would you need to know, beyond just the footprints, to precisely reconstruct your entire journey from start to finish? (Think about any missing information that differentiation "loses".) (Think about the "starting point" or any fixed reference.) the Fading Footprints Value Integration: Antiderivatives Connection: Finding antiderivatives is like retracing the steps of a function back to its source. In calculus, this process allows engineers and scientists to recover original quantities from rates of change —an essential tool for solving problems in physics, engineering, and beyond. "Precept upon precept, line upon line…" — Isaiah 28:10 Inlife, building understanding is also a step- by-step journey. Just as each small piece of an integral adds up to a whole, our daily choices, habits, and reflections shape our character and wisdom over time. By patiently piecing together knowledge and applying it faithfully, we honor God’s design for growth and stewardship of our gifts.

DISCUSSION: The Inverse of Differentiation Defining the Antiderivative An antiderivative is the reverse operation of differentiation. If we have a function f(x), its antiderivative is a function F(x) such that the derivative of F(x) is f(x). In other words, if F′(x)=f(x), then F(x) is an antiderivative of f(x). Example: We know that the derivative of x^2 is 2x. So, x^2 is an antiderivative of 2x. We also know that the derivative of x^2 +5 is 2x. So, x^2 +5 is also an antiderivative of 2x. And the derivative of x^2 -100 is 2x. So, x^2 -100 is yet another antiderivative of 2x. This observation leads us to a crucial point: Did you know that?In 1666, while staying at home during a plague outbreak, Sir Isaac Newton developed the foundational ideas of both derivatives and antiderivatives, which later became key parts of calculus. He realized that finding an area under a curve could be simplified by finding a function whose derivative is the given curve—what we now call an antiderivative. This insight led to the creation of the Fundamental Theorem of Calculus, bridging the processes of differentiation and integration. Thanks to this discovery, engineers and scientists can now easily solve problems involving motion, growth, and energy. For example, they can determine the total distance an object travels from its speed or find the original population from a known growth rate. This powerful mathematical tool, born out of a time of crisis, still helps solve real-world challenges today—from predicting how buildings respond to stress to analyzing electrical circuits and even modeling how data flows through networks.Understanding the Constant of Integration (C) Notice that in the examples above, any constant added to x2 still results in 2x when differentiated. This is because the derivative of any constant is zero. Therefore, if F(x) is an antiderivative of f(x), then F(x)+C (where C is any arbitrary real number constant) is also an antiderivative of f(x). This 'C' is called the Constant of Integration. 03 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

Introduction to the Indefinite Integral Notation The process of finding an antiderivative is called anti-differentiation or integration. The symbol used to denote the general antiderivative (or indefinite integral) of a function f(x) is the integral sign: ∫f(x)dx This expression is read as "the indefinite integral of f(x) with respect to x." ∫ is the integral sign. f(x) is the integrand (the function to be integrated). dx indicates that we are integrating with respect to the variable x. So, the general form of an indefinite integral is: ∫f(x)dx=F(x)+C where F′(x)=f(x) and C is the constant of integration.When we find the antiderivative of a function, we are doing the opposite of taking a derivative. The important thing to remember is: There are always many possible antiderivatives! This is because you can add or subtract any constant, and it won’t change the derivative. Finding an antiderivative means finding the original function plus a constant. So we always add + C at the end! INTEGRATION AND THE INDEFINITE INTEGRAL Integration and antidifferentiation mean the same thing — they both mean finding the antiderivative of a function. We use the integral symbol ∫ to show this process. When finding antiderivatives, we often use the power rule for integration: Key Terms ✅ ∫ — this is called the integral symbol. ✅ f(x) — this is called the integrand (the function you are integrating). ✅ dx— shows the variable of integration. ✅ Indefinite integral — means we don’t have limits (just the general antiderivative with +C). This means:1️⃣ Add 1 to the exponent.2️⃣ Divide by the new exponent.3️⃣ Don’t forget + C! Key Tip: ✅ For the power rule, never use it for n=−1 For , use instead. RATIONAL FUNCTIONSRATIONAL FUNCTIONS IRRATIONAL FUNCTIONSIRRATIONAL FUNCTIONS FELLOWSHIP BAPTIST COLLEGE 04fellowship baptist college 02

TRIGONOMETRIC FUNCTIONSTRIGONOMETRIC FUNCTIONS INVERSE TRIGONOMETRIC FUNCTIONSINVERSE TRIGONOMETRIC FUNCTIONS EXPONENTIAL FUNCTIONSEXPONENTIAL FUNCTIONS HYPERBOLIC FUNCTIONSHYPERBOLIC FUNCTIONS LOGARITHMIC FUNCTIONSLOGARITHMIC FUNCTIONS SPECIAL FUNCTIONSSPECIAL FUNCTIONS Did you know that? Before the integral sign was invented, mathematicians used Σ (sigma) notation for Riemann sums, which approximate the area under a curve. The integral sign eventually evolved as a continuous counterpart to this discrete sum. The integral sign "∫" was introduced by Gottfried Wilhelm Leibniz in the late 1600s. It comes from the elongated Latin letter "S", which stands for “summa” or “sum” representing the summing of infinitely small parts. 05 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

PROBLEM01SOLUTION:✅ How to Integrate Functions with Multiple Terms When you see an integral with more than one term, follow these steps:📌 Steps to Follow1️⃣ Factor out constants If there are numbers multiplying the terms, take them outside the integral. Example:2️⃣ Use the sum and difference rules You can split the integral for each term, using:3️⃣ Apply the power rule (or other rules) to each term Use the antiderivative formulas for each part.Always add + C at the end for the general solution!PROBLEM02SOLUTION:FELLOWSHIP BAPTIST COLLEGE 06

All four functions contain only one term. Each term contains a base of x and an exponent, so we’ll use the power rule for integration Find the antiderivative by increasing the exponent by 1. Divide the resulting expression by new exponent of x. Include a “+C” after the antiderivative to cover all functions belonging in the same family.PROBLEM03SOLUTION:PROBLEM04SOLUTION:PROBLEM05SOLUTION:07 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

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KEY PRINCIPLES TO MASTER Understand the Concept of Antiderivatives Grasp how taking an antiderivative is the reverse process of differentiation. Use the Indefinite Integral Symbol (∫) Recognize and properly write antiderivatives using the integral sign, adding + C. Break Down Complex Expressions Use sum and difference rules to integrate functions term by term. Mastering these principles builds the foundation for solving definite integrals, applying advanced techniques, and tackling engineering problems with confidence!1️⃣ Antiderivatives are used to find the original function when we know how it changes. It’s like working backwards from the derivative.2️⃣ Indefinite integrals are another name for antiderivatives. They always include an extra constant at the end because there are many possible original functions that differ only by a constant value.3️⃣ The power rule for integration says you increase the exponent by one and then divide by the new exponent. This is the opposite of the power rule for derivatives.4️⃣ When integrating functions with multiple terms, first take out any number multiplying the terms to make the calculation easier.5️⃣ Next, split the expression into separate parts for each term, and then apply the correct integration rule to each part.6️⃣ Finally, combine all the results and always remember to add the constant at the end. In our study of integration, we discover meaningful parallels to how we piece together life’s experiences into a greater whole. Just as integration combines small parts to find total area or accumulated change, we too are called to see the bigger picture of our actions, choices, and purpose. Understanding antiderivatives reminds us that every step we take builds upon the past, shaping the present and future. Just as engineers use integration to solve real-world problems, we are encouraged to apply wisdom and diligence in uniting knowledge, faith, and discipline to solve life’s challenges. By mastering these foundational concepts, we not only strengthen our mathematical skills but also grow in our ability to live purposefully and honor God through our work and understanding. Stewart, J. (2016). Calculus: Early transcendentals (8th ed.). Cengage Learning. Rohde, A., Jain, S. K., & Poddar, A. (2012). Introduction to integral calculus: Systematic studies with engineering applications for beginners. Wiley. Boas, M. L. (2005). Mathematical methods in the physical sciences (3rd ed.). Wiley. Thomas, G. B., Weir, M. D., & Hass, J. (2018). Thomas’ calculus (14th ed.). Pearson.REFERENCES SUMMARY: 09 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

FELLOWSHIP BAPTIST COLLEGE 10DEFINITE AND IMPROPERLEARNING OBJECTIVESBy the end of this module, students should be able to:1️⃣ Understand and apply advanced integration techniques — including substitution, integration by parts, partial fractions, and methods for trigonometric, exponential, hyperbolic, and inverse trigonometric functions — to solve complex integrals.2️⃣ Recognize and evaluate improper integrals, analyzing when and how to handle integrals with infinite limits or unbounded behavior to ensure valid solutions.3️⃣ Develop problem-solving strategies by selecting appropriate integration methods for different types of functions, verifying results, and interpreting solutions in mathematical and real-world contexts.4️⃣ Appreciate the role of integration in engineering, science, and technology, recognizing how these advanced techniques help model, analyze, and solve problems across various fields. As engineering students, it is easy to view calculus as merely a toolkit of procedures and formulas. However, when we look closer, the mechanics of integration offer a beautiful reflection of God’s character and His relationship with creation. In Definite Integrals, we see the beauty of God’s order and boundaries. He is a God who numbers the hairs on our heads and sets precise physical laws to govern the universe. Just as a definite integral perfectly sums up millions of microscopic pieces to find an exact, orderly total, God orchestrates the small, seemingly disconnected moments of our daily lives, integrating them into a complete and purposeful design. In Improper Integrals, we encounter a shadow of God’s transcendence. Human minds struggle to comprehend the concept of eternity or a God without limits. Yet, improper integrals give us a precise mathematical language to approach the infinite. They remind us that even when a situation looks boundless, unpredictable, or stretched out into an infinite horizon, there is still an underlying truth, a resolution, and a defined value holding it together. The fact that a shape can stretch out infinitely far yet possess a finite, stable area mirrors how God’s limitless, infinite grace can break into our small, finite human lives and bring us ultimate wholeness. As you work through every boundary, limit, and convergence test in this module, let it remind you that you are studying the language of a Creator who is both intimately involved in the finest, defined details of your life, and sovereign over the boundless expanse of eternity. approach these problems not just with mathematical grit, but with a heart of wonder! INTEGRAL

The concept of improper integrals has deep historical roots, tracing back to the 17th and 18th centuries when early mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz laid the groundwork for calculus. However, it was the brilliant mathematician Leonhard Euler who significantly advanced the understanding of infinite series and integrals that extend to infinity. Euler and later mathematicians such as Augustin-Louis Cauchy formalized how to handle integrals with infinite limits or unbounded functions — ideas that seemed paradoxical at the time. Their work paved the way for modern probability, physics, and engineering calculations that deal with infinite intervals and singularities, proving that “improper” doesn’t mean “incorrect” but rather “beyond the standard limits” of ordinary integration. LESSON OVERVIEW: Understanding improper integrals is essential in advanced integral calculus, as they extend the concept of definite integrals to cases where the limits of integration are infinite or the function becomes unbounded within the interval. By learning how to handle these special cases, students gain tools to evaluate areas and accumulated quantities that cannot be addressed using standard techniques—crucial in fields such as engineering, physics, and probability theory. Improper integrals allow for precise calculation of quantities like infinite series, electric fields, or probabilities over unbounded domains. Mastering improper integrals also plays a vital role in expanding mathematical and engineering problem-solving capabilities. It supports the development of advanced models, enables the solution of problems involving unbounded behavior, and underpins integral convergence tests used in analysis. Whether applied in signal processing, thermodynamics, or structural engineering, understanding how to evaluate improper integrals ensures accurate interpretation of infinite or extreme scenarios. As students progress in calculus, a solid grasp of improper integrals prepares them for deeper studies in analysis, differential equations, and real-world applications involving infinite processes and limits. VALUES INTEGRATIONDID YOU KNOW THAT? Trusting Beyond Limits "Trust in the Lord with all your heart and lean not on your own understanding." — Proverbs 3:5 Improper Integrals Connection: Improper integrals allow us to handle situations where a function stretches out infinitely or behaves unpredictably at certain points. This concept equips engineers and scientists to analyze real-world phenomena that seem boundless or undefined, such as infinite series, wave behavior, or probability over infinite ranges. Value Integration: In life, we too face moments that feel unbounded or difficult to grasp fully. Just as we rely on special techniques to make sense of improper integrals, we are called to trust God’s infinite wisdom when our own understanding falls short. By acknowledging our limits and leaning on His guidance, we navigate uncertainties with faith, humility, and hope, knowing that His knowledge has no bounds. IMPROPER INTEGRALIMPROPER INTEGRAL What Are Improper Integrals? Improper integrals are a special type of definite integral distinguished by their limits or behavior. Unlike regular definite integrals, which are evaluated over finite intervals with a bounded integrand, improper integrals may extend to infinity or involve an unbounded function within the interval. Key Characteristics: ✅ Infinite Limits: An improper integral has at least one limit of integration that is infinite (e.g., ). ✅ Unbounded Integrand: Alternatively, the function being integrated might become infinite at some point within the interval. 11 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

How to Tell If an Integral is Improper A definite integral is improper if any of these conditions apply:1️⃣ One or both limits of integration are infinite2️⃣ The integrand is unbounded within the intervalTypes of Improper Integrals ✅ 1) Improper Integrals with One Infinite Limit Form: How to evaluate: Replace the infinite limit with a variable, then take the limit: Convergent if this limit is finite. Divergent if it is infinite or does not exist.✅ 2) Improper Integrals with Two Infinite Limits Form: How to evaluate: Split into two improper integrals with one infinite limit each: The whole integral is convergent only if both parts converge. ✅ Key Point An improper integral converges if its limit(s) produce a finite value. Otherwise, it diverges. DISCUSSION IMPROPER INTEGRALDoes It Converge or Diverge? An improper integral converges if its evaluation results in a finite value. If not, it diverges.Why Are They Important? Improper integrals help us solve problems involving unbounded domains or singularities — for example, in probability, physics, and engineering.✅ 3) Improper Integrals with Discontinuities Where: The function f(x) has a discontinuity at an endpoint or inside the interval. Convergent if both parts are finite. The definite integral, a core concept in calculus, has roots dating back to ancient times! The method of exhaustion—an early technique for finding the area under a curve—was used by Greek mathematicians like Archimedes around 250 BC. He cleverly approximated the area of a circle by inscribing polygons with more and more sides, laying the groundwork for modern integral calculus. Today, definite integrals are vital tools for engineers: they help calculate quantities like total current, energy consumption, or signal power in electronic systems. So, every time you use an app, watch a video, or charge a device, you’re benefiting from centuries-old mathematical ideas refined into powerful modern tools!DID YOU KNOW THAT? FELLOWSHIP BAPTIST COLLEGE 12

The definite integral's ability to sum up infinitesimal parts to yield a precise, finite whole beautifully mirrors the intricate and purposeful nature of God's creation. Each "slice" of a phenomenon, no matter how small, contributes meaningfully to the overall design and outcome. This concept encourages us to recognize the significance of incremental progress and the faithfulness of divine design, where every detail matters in achieving a larger purpose. 13The definite integral is one of the most powerful and widely used tools in calculus, providing a rigorous way to calculate the net accumulation of a quantity that varies continuously. While indefinite integrals (antiderivatives) give us a family of functions, definite integrals yield a single, numerical value representing a total quantity over a specific interval. This numerical value can represent diverse physical concepts such as area, displacement, total work, volume, and many more, DEFINITE INTEGRALDEFINITE INTEGRALThe Fundamental Theorem of Calculus, (The Evaluation Theorem) While the definition of the definite integral involves a complex limit of sums, the Fundamental Theorem of Calculus provides a remarkably straightforward method for its evaluation, linking it directly to antiderivatives. Theorem: If f is continuous on [a,b] and F is any antiderivative of f (i.e., then: Significance: This theorem transforms the challenging problem of finding a limit of sums into the relatively simpler problem of finding an antiderivative and evaluating it at two points. This is the primary method used to compute definite integrals in practice.PROBLEM01SOLUTION: COLLEGE OF ENGINEERING AND COMPUTER STUDIES

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INTEGRATION BY PARTSINTEGRATION BY PARTS✅ What is Integration by Parts? Integration by parts uses the product rule for derivatives in reverse. If you have: then you can tackle an integral by choosing parts of the integrand to be u and dv. The goal is to end up with a new integral, , that’s easier to solve than the original.How to Use Integration by Parts Steps:1️⃣ Identify: Rewrite the integral as a product .2️⃣ Choose: Pick u and dv: u: Easy to differentiate dv: Easy to integrateUse LIATE as a guide (priority for choosing u): L — Logarithmic I — Inverse trig A — Algebraic (like x) T — Trigonometric E — Exponential3️⃣ Compute: Differentiate u → find du Integrate dv → find v4️⃣ Apply:5️⃣ Simplify: Solve the new integral and combine terms.When to Use Integration by Parts Use IBP when: The integrand is clearly a product of two different kinds of functions (e.g., polynomial × exponential, polynomial × trig, log × algebraic). Direct integration is complicated, but the derivative or antiderivative of each part is simpler. Remember: Good choices for u make the resulting integral easier than the original.Key Tip Integration by parts may sometimes need to be repeated more than once — especially for polynomials multiplied by trig or exponential functions.EXAMPLE13SOLUTION: 17 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

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Algebraic long division and partial fraction decomposition are fundamental analytical tools in mathematics and engineering, enabling the transformation of complex, unmanageable rational expressions into simpler components that are easily integrated. Rational expressions are broadly classified based on the degrees of their polynomial numerators and denominators. When the degree of the numerator is greater than or equal to the denominator, algebraic long division is used to reduce the expression. For instance, in control systems and signal processing, transfer functions must often be broken down to evaluate system stability and time response. Each integration technique exhibits predictable, rule-governed steps that determine how smoothly an engineered system can be modeled and analyzed. Understanding these algebraic behaviors is essential in advanced engineering and physics applications. Long division, decomposition, and integration are fundamental processes that explain how dynamic variables interact within a system. For example, electrical circuit analysis relies on partial fractions to invert Laplace transforms, while chemical engineering uses these techniques to determine the reaction rates of complex mixtures. These mathematical laws underpin structural structural modeling, fluid mechanics, and automated control systems. From a Christian perspective, studying the systematic breakdown of complex expressions reflects the intricate orderliness and logic of God’s creation. The precise behavior of mathematical laws demonstrates the consistent principles He established. As engineers, applying this knowledge responsibly aligns with stewardship, diligence, and care for others (Colossians 3:23). Knowledge of advanced integration empowers students to develop technologies that serve humanity ethically and effectively, whether in infrastructure development, medical technology, or environmental systems. ALGEBRAIC LONG DIVISION &LEARNING OBJECTIVESBy the end of this module, students should be able to: Analyze the structural differences between proper and improper rational functions, determining when to apply algebraic long division before integration. Evaluate complex integrals by decomposing proper rational functions into distinct linear, repeated linear, or irreducible quadratic partial fractions to predict system behaviors. Design mathematical models or solutions demonstrating how decomposing complex systems into simpler components solves real-world engineering problems, such as circuit analysis or fluid dynamics. Apply Christian stewardship by demonstrating diligence, precision, and intellectual honesty when breaking down complex technical problems into manageable, ethical solutions for societal benefit.PARTIAL FRACTION INTEGRATION FELLOWSHIP BAPTIST COLLEGE 24

LESSON OVERVIEW: This lesson introduces the fundamental concepts of algebraic long division and partial fraction decomposition as prerequisite tools for integration. Students will explore practical applications in electronics and mechanical engineering, such as modeling circuit impedance, analyzing fluid flow rates, and solving differential equations, with examples reflecting local infrastructure challenges. By analyzing polynomial behavior in different mathematical environments, students develop the ability to predict system performance, design efficient analytical pathways, and solve real-world engineering problems. Understanding the precision and predictability of mathematical laws reflects the order and wisdom inherent in God’s creation. By applying this knowledge ethically, engineers act as stewards of technology, ensuring that innovations enhance human life, protect communities, and serve society responsibly. This perspective encourages students to integrate diligence, integrity, and care for others into their engineering practice, aligning technical competence with Christian values of stewardship and service ($Colossians\ 3:23$). CHRISTIAN PERSPECTIVE:MOTIVATIONAL ACTIVITY "The Algebraic Readiness Check" Task 1: The Degree Check (Concept Reading) Review your textbook notes on polynomial functions. In your own words, define what makes a rational expression P(x)}/Q(x) proper versus improper based on its degree. Task 2: Structural Classification Analyze the four mathematical structures below. For each expression, simply state whether your first algebraic step must be Algebraic Long Division or Partial Fraction Decomposition: Instructions: Complete this task individually in your study notebooks before coming to our next class session. Your responses will serve as your entry pass and foundation for the opening motivational activity. DISCUSSION I. PREREQUISITE KNOWLEDGE Before studying Partial Fraction Integration, students must know the following: Algebra Skills Students must be able to: ✓ Factor polynomials 25 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

THE BIG IDEA OF PARTIAL FRACTIONS Main Concept Some rational functions are difficult to integrate directly. The solution is to break one complicated fraction into several simpler fractions. Instead of integrating directly, we rewrite it as that we already know how to integrate. Think of it as: Complicated fraction → Smaller fractions → Easy integration ✓ Recognize irreducible quadratics An irreducible quadratic cannot be factored over the real numbers. Therefore, it cannot be factored. Degree of a Polynomial The degree is the highest exponent. IMPORTANT RULE Never apply partial fractions immediately to an improper rational function. You MUST first perform long division. INTEGRATION USING LONG DIVISION AND PARTIAL FRACTIONSINTEGRATION USING LONG DIVISION AND PARTIAL FRACTIONS1️⃣ When to Use Long Division Sometimes, you need to integrate a rational function — a fraction where both numerator and denominator are polynomials. If the degree of the numerator ≥ the degree of the denominator, use polynomial long division first. Why? Long division rewrites the fraction into a polynomial plus a proper fraction (where the numerator’s degree is smaller than the denominator’s). This makes integration much easier. FELLOWSHIP BAPTIST COLLEGE 26

PROBLEM01SOLUTION:PROBLEM03SOLUTION:PROBLEM02SOLUTION:PROBLEM04SOLUTION:Practice ProblemsPerform Integration if the following integrands are given 27 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

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FELLOWSHIP BAPTIST COLLEGE 32 PARTIAL FRACTION A rational algebraic fraction is an expression of the form , where 𝑢 and 𝑣 are polynomials and . If the degree of the numerator is less than the degree of the denominator, it is called a proper fraction. To break a given rational fraction into partial fractions, first factor the denominator completely into real factors — these factors may be linear, irreducible quadratic, or repeated. Once factored, we can decompose the proper fraction into a sum of partial fractions using four possible cases. For example, when adding two fractions in algebra, we find the least common denominator (LCD), divide the LCD by each denominator, multiply by the corresponding numerator, then combine and simplify. In partial fractions, we do the reverse: given a single fraction, we split it into simpler fractions whose denominators are the factors of the original denominator. In this section, instead of adding fractions, we will do the opposite: we will break down a single fraction into a sum of simpler fractions — this process is called partial fraction decomposition. We apply partial fraction decomposition to rational functions of the form: where and are polynomials. Before starting, remember these important steps: Make sure the degree of (the numerator) is less than the degree of q(x) (the denominator). If it is not, use long division first to rewrite the fraction. Next, factor completely into linear factors and/or irreducible quadratic factors. Once factored, we use the following rules to write the partial fraction decomposition, depending on the types of factors found. a.Linear factors. Linear factors are of the form mx + b.

33 COMMON MISTAKES TO AVOIDCOMMON MISTAKES TO AVOIDPROBLEM01 INTEGRATION FORMULAS TO MEMORIZEINTEGRATION FORMULAS TO MEMORIZE ❌ Skipping long division. ❌ Forgetting to factor completely. ❌ Using constants A,B over quadratic denominators instead of Ax+B. ❌ Forgetting repeated factors. ❌ Integrating before solving for constants. ❌ Forgetting +C. COLLEGE OF ENGINEERING AND COMPUTER STUDIES

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Integration using partial fractions is a technique used to evaluate rational functions by rewriting a complicated fraction into simpler fractions that can be integrated using basic integration rules. The process begins by determining whether the rational function is proper or improper; if it is improper, algebraic long division must be performed first. Next, the denominator is factored completely to identify the appropriate type of partial fraction decomposition, such as distinct linear factors, repeated linear factors, irreducible quadratic factors, or repeated irreducible quadratic factors. After setting up the decomposition and solving for the unknown constants, the expression is rewritten into simpler terms and integrated using standard formulas such as the power rule, logarithmic rule, and inverse trigonometric rule. This systematic approach transforms complex rational expressions into manageable integrals and is widely used in calculus, engineering, and applied mathematics. Stewart, J. (2016). Calculus: Early Transcendentals (8th ed.). Cengage Learning; Thomas, G. B., Weir, M. D., & Hass, J. (2018). Thomas' Calculus (14th ed.). Pearson; Advanced Engineering Mathematics. Kreyszig, E. (2011). Advanced Engineering Mathematics (10th ed.). Wiley; The Holy Bible, 1 Corinthians 14:33. SUMMARY: From a Christian perspective, studying mathematics reflects God's nature as a God of order, wisdom, and design (1 Corinthians 14:33), reminding us that creation follows patterns and principles that can be understood and used responsibly to serve others and contribute to society.REFERENCES 43 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

INTEGRATIONLEARNING OBJECTIVESBy the end of this module, students should be able to:Identify appropriate inner functions u and corresponding differentials du within composite integrands. Apply the Change of Variables (u-substitution) method to transform complex integrals into standard forms. Evaluate integrals involving composite, exponential, trigonometric, and algebraic functions using systematic substitution. Translate non-linear expressions into simplified structures suitable for integration. Demonstrate discipline, accuracy, and intellectual integrity in mathematical problem-solving as an expression of Christian stewardship. Integrating composite functions of the form that follows u substitution is a fundamental process in calculus and engineering, enabling the transformation of complex, interdependent rates into predictable equations. Integrands are broadly classified into basic and composite functions based on their inner structural nesting. When an expression contains an "inner function" multiplied by a scalar multiple of its derivative, standard power or geometric rules cannot be applied directly. In these cases, u-substitution (the reverse Chain Rule) must be performed to shift the mathematical variable from x to u, streamlining the expression into a standard remainder. Once a simplified proper differential form is obtained, direct antiderivative evaluation is used to solve the integration before mapping the final result back into original system variables. Understanding these structural substitutions is essential in advanced engineering applications. Identifying functions and their derivatives is a rule-governed process that allows us to isolate dependent components within a complex system. For instance, in electrical circuit analysis, resolving transient responses in circuits with varying inductive fields requires variable substitution to integrate exponential current functions over time. Similarly, in structural mechanics, tracking stress distribution gradients across irregularly curved cross-sections requires substitution methods to integrate complex polynomial geometry. These precise integration methods ensure that even the most formidable mathematical structures can be solved systematically. From a Christian perspective, studying the absolute and consistent laws of integration reflects the intricate orderliness and logic of God’s creation. The precise behavior of mathematical equations demonstrates the fixed principles He established within the universe. As engineering students, mastering these rigorous calculus methods teaches us academic diligence and intellectual honesty. Transforming complex, un-integrable expressions into orderly, verifiable solutions trains the mind to approach large-scale problems with patience, aligning technical competence with stewardship and a commitment to excellence (Colossians 3:23).USING U-SUBSTITUTION 51 FELLOWSHIP BAPTIST COLLEGE 44

DID YOU KNOW THAT? U-substitution is widely used in physics, especially in solving problems involving exponential decay like radioactive decay and capacitor discharge in electrical circuits? Without u-substitution, many real-world models (such as growth, motion, and heat transfer equations) would be extremely difficult or impossible to solve using basic antiderivative rules alone? Mathematicians often describe u-substitution as a method of “changing perspective”—you are not changing the problem, but changing the lens through which you view it. 17 LESSON OVERVIEW: This lesson focuses on the systematic use of u-substitution to evaluate integrals involving composite functions. Students will: Analyze nested function structures Identify appropriate substitution strategies Perform algebraic transformations using differentials Evaluate integrals in simplified form Interpret results in applied engineering contexts The goal is to develop both procedural fluency and structural understanding. The precision and order found in calculus reflect the consistency and intelligibility of God’s creation. Mathematical laws operate with structure and reliability, pointing to a universe governed by design rather than randomness. As students develop skill in transforming complexity into clarity, they practice discipline, patience, and intellectual honesty—values aligned with Christian stewardship. “Whatever you do, work at it with all your heart…” — Colossians 3:23 In this sense, mathematical learning becomes not only technical training but also formation in character and responsibility. CHRISTIAN PERSPECTIVE:MOTIVATIONAL ACTIVITY "Integration Puzzle: Tracing Hidden Structure"Objective: To strengthen analytical reasoning by identifying substitution structures in composite integrals and connecting mathematical thinking to collaborative problem-solving. Instructions: Activity: Students will form groups of four and analyze the following integrals: Group Tasks: 1. Structural Identification Identify the correct substitution: Choose u Determine du Justify why the structure matches u-substitution 2. Transformation Blueprint Rewrite each integral conceptually in u-form (no final answers required), showing how the expression simplifies after substitution. 3. Presentation Each group will present: Their substitution strategy The reasoning behind selecting uuu How recognizing structure simplifies the problem 45 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

INTEGRATION BY U SUBSTITUTIONINTEGRATION BY U SUBSTITUTIONWhat is U-Substitution? U-substitution is a basic but powerful method for finding antiderivatives. It is essentially the reverse of the chain rule in differentiation. This technique allows us to simplify an integral by changing variables — we substitute a complicated expression with a single variable, usually u.When to Use U-Substitution Use u-substitution when: The integrand contains a function and its derivative. You can spot an inner function whose derivative is present elsewhere in the integrand. It works best when you see an expression inside another function (like inside a power, root, or trig function) and its derivative is multiplying it.How to Apply U-Substitution ✅ Steps1️⃣ Choose u: Identify the inner function and let 2️⃣ Find du: Differentiate u with respect to x:3️⃣ Rewrite the integral: Replace all parts in terms of u and du.4️⃣ Integrate: Solve the simpler integral in terms of u.5️⃣ Back-substitute: Replace u with your original expression to write the answer in terms of x.Definite Integrals with U-Substitution When solving definite integrals, you have two options: 1.Change the limits to match u: If x=a, find u=g(a); if x=b, find u=g(b). 2.Keep the original limits but plug back in x after integrating.Key Idea Think of u-substitution as changing the ‘language’ of the integral to make it easier to solve — just like the chain rule ‘hides’ an inside function when differentiating.PROBLEM01SOLUTION:PROBLEM02SOLUTION: FELLOWSHIP BAPTIST COLLEGE 46

STRUCTURAL UNDERSTANDING U-substitution works because it recognizes hidden derivative pairs inside integrals. A correct substitution is usually identified when: A function is inside another function (composite structure) Its derivative (or a constant multiple) is also present in the integrand This method transforms complex expressions into standard forms such as: Power rule Exponential rule Trigonometric integrals APPLICATION IN ENGINEERING In engineering systems, u-substitution is used to simplify real-world models such as: Electrical circuits involving exponential growth and decay Mechanical systems with variable acceleration Thermal systems involving heat transfer equations By reducing complex expressions into standard forms, engineers can analyze systems more efficiently and accurately. KEY REMINDER Successful u-substitution depends on pattern recognition, not memorization. Always ask: “Is there a function inside another function, and is its derivative also present?” If yes, u-substitution is likely the correct method.PROBLEM03SOLUTION:PROBLEM04SOLUTION: 47 COLLEGE OF ENGINEERING AND COMPUTER STUDIES

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FELLOWSHIP BAPTIST COLLEGE 54 This module on u-substitution teaches students how to simplify complex integrals by identifying hidden inner functions and their derivatives, then transforming them into standard forms through a change of variables. By recognizing structures such as logarithmic, exponential, trigonometric, and hyperbolic relationships, learners develop the ability to systematically reduce complicated expressions into solvable integrals. This process strengthens analytical thinking and problem- solving skills that are widely applied in engineering and science, where modeling real-world systems often requires converting intricate relationships into manageable mathematical forms. Stewart, J. (2016). Calculus: Early Transcendentals (8th ed.). Cengage Learning; Thomas, G. B., Weir, M. D., & Hass, J. (2018). Thomas' Calculus (14th ed.). Pearson; Advanced Engineering Mathematics. Kreyszig, E. (2011). Advanced Engineering Mathematics (10th ed.). Wiley; The Holy Bible, 1 Corinthians 14:33. SUMMARY: From a Christian perspective, the order and consistency found in these mathematical laws reflect the reliability and intelligibility of God’s creation, encouraging students to approach their studies with diligence, integrity, and stewardship as taught in Colossians 3:23. The discipline required in mastering substitution mirrors a broader commitment to faithful and excellent work in all intellectual pursuits.REFERENCES

COLLEGE OF ENGINEERING AND COMPUTER STUDIES 55 Trigonometric functions are fundamental mathematical models used to describe periodic and oscillatory behavior in engineering, physics, and many natural phenomena. Integrals involving powers and products of sine and cosine functions frequently appear when analyzing wave motion, alternating current circuits, signal processing, vibration systems, and harmonic oscillations. Unlike basic algebraic functions, trigonometric expressions often require strategic transformations before direct integration can be performed. Integrands involving sine and cosine powers follow a structured classification system. If one exponent is odd, a trigonometric factor is isolated and the remaining expression is transformed using Pythagorean identities. If both exponents are even, half-angle identities are applied to reduce the expression into simpler forms. When sine and cosine functions have different angles, product-to-sum identities convert products into simpler trigonometric sums that can be integrated directly. This systematic approach transforms complicated expressions into manageable mathematical forms. Understanding these transformations is essential in engineering applications. In electrical engineering, alternating current systems are represented by sinusoidal functions whose energy outputs require trigonometric integration. In mechanical engineering, vibration analysis and harmonic motion depend on trigonometric models to predict displacement and energy distribution over time. In civil engineering, periodic loading and resonance analysis also rely on these mathematical tools. These methods allow engineers to simplify and accurately model real-world systems. From a Christian perspective, studying the order and consistency of trigonometric identities reflects the wisdom and structure God established within creation. The elegant relationships among mathematical functions demonstrate that the universe operates according to dependable principles. As engineering students, mastering these systematic methods cultivates patience, discipline, and intellectual honesty. Learning to uncover hidden patterns within complex mathematical structures develops stewardship and a commitment to excellence in solving societal problems (Colossians 3:23).INTEGRATION OF TRIGONOMETRIC FUNCTIONSLEARNING OBJECTIVESBy the end of this module, students should be able to: Analyze trigonometric integrands involving powers and products of sine and cosine functions and classify them according to the appropriate integration strategy. Evaluate trigonometric integrals by applying trigonometric identities, half- angle formulas, and product-to-sum identities to transform complex expressions into standard antiderivative forms. Design systematic algebraic procedures for solving integrals involving odd powers, even powers, and products of trigonometric functions with different angles. Apply Christian stewardship by demonstrating diligence, precision, and intellectual honesty when uncovering hidden mathematical patterns that contribute to engineering and societal development.

FELLOWSHIP BAPTIST COLLEGE 56 LESSON OVERVIEW: This lesson focuses on the mathematical principles governing the integration of powers and products of sine and cosine functions. Students will analyze integrands, classify them according to specific cases, apply trigonometric identities, and systematically simplify expressions before integration. Through these processes, students will develop analytical reasoning, algebraic fluency, and procedural accuracy necessary for advanced engineering applications. Understanding the symmetry and predictability of trigonometric identities reflects the wisdom and consistency inherent in God's creation. By practicing careful analysis and precise calculations, students cultivate discipline, integrity, and stewardship that extend beyond mathematics into ethical engineering practice. Pursuing excellence in mathematical problem-solving honors God through diligent and responsible use of knowledge (Colossians 3:23). CHRISTIAN PERSPECTIVE:MOTIVATIONAL ACTIVITY "The Trigonometric Strategy Challenge" Objective: To develop analytical reasoning by classifying trigonometric integrals before solving them while connecting mathematical organization to collaborative engineering thinking. Instructions: Activity: Students will form groups of four and evaluate the following unsolved integrals displayed on the board. Group Tasks 1. The Structural Sorting Determine which integration strategy applies to each integral. 2. The Identity Blueprint List all trigonometric identities required before integration begins. 3. The Strategy Mapping Construct a flowchart showing the sequence of steps needed to simplify each integral before integrating. Example: Sine Odd Identify odd sine → Save one sine → Convert remaining sine powers → Let u=cosx → Integrate Both Even Apply half-angle identities → Simplify → Integrate Different Angles Apply product-to-sum identity → Simplify → Integrate DISCUSSIONTrigonometric functions are mathematical models used to describe periodic and oscillatory phenomena. They appear in many engineering applications such as alternating current circuits, wave motion, signal processing, sound analysis, structural vibrations, and harmonic motion. Unlike basic polynomial functions, trigonometric functions cannot always be integrated directly. Many trigonometric integrals require the use of identities and algebraic transformations before standard integration rules can be applied. The key to solving these integrals is pattern recognition. Students must first classify the integrand and then apply the appropriate strategy.There are four major cases when integrating sine and cosine functions: Sine exponent is odd Cosine exponent is odd Both exponents are even Sine and cosine have different angles (product-to- sum)

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FELLOWSHIP BAPTIST COLLEGE 60ENGINEERING CONNECTION Trigonometric integrals are used in: Alternating current circuits Signal processing Wave mechanics Vibration analysis Harmonic motion Structural resonance Telecommunications Engineers use these mathematical tools to predict cyclic behavior, energy transfer, and periodic motion accurately. CHRISTIAN REFLECTION The elegant relationships among trigonometric identities reflect the order and consistency embedded within God's creation. Every transformation follows a dependable pattern, reminding us that complexity can often be resolved through patience, discipline, and careful analysis. As future engineers, our responsibility is to apply these principles with integrity and excellence, honoring God through diligent stewardship of knowledge and service to society. “Whatever you do, work at it with all your heart, as working for the Lord, not for human masters.” (Colossians 3:23) Did you know that? Trigonometric Integrals Help Power Your Home The electricity delivered to homes and buildings is called alternating current (AC) because it changes direction periodically in a sinusoidal pattern. Engineers use sine and cosine functions to model these electrical signals, and trigonometric integration helps determine energy consumption, average power, and circuit performance. Mathematics Reflects Order in Creation The predictable relationships among trigonometric functions demonstrate that the universe operates according to consistent and dependable principles. From a Christian perspective, this orderliness reflects God's wisdom and encourages us to approach our studies with diligence, discipline, and stewardship. “For God is not a God of disorder but of peace.” (1 Corinthians 14:33)

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TAN–SEC AND COT–CSC CASES Integrals involving tangent and secant functions, as well as cotangent and cosecant functions, also follow a pattern-recognition approach. Unlike sine and cosine integrals, these functions possess derivative relationships that make certain combinations easier to integrate after strategic algebraic manipulation. The key to solving these integrals is identifying whether an integrand contains enough factors that can produce a derivative pair. Students should always remember these derivative relationships. FELLOWSHIP BAPTIST COLLEGE 66

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Shaping Lives.CALCULUS 2: INTEGRAL CALCULUSCALCULUS 2: INTEGRAL CALCULUSENGMATH 122ENGMATH 122