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13.2 Integration by Substitution

This guide walks you through the process of integration by substitution, emphasizing its significance in calculus. Starting with the inner function, you will learn the steps of cross-multiplying, substituting into the original integral, and ultimately taking the integral to express the final answer in terms of x. This method not only simplifies complex integrals but also serves as a foundation for understanding calculus concepts such as derivatives and their applications in real-world scenarios, including population growth and optimization.

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13.2 Integration by Substitution

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  1. 13.2 Integration by Substitution

  2. Let w be the inside function • Write • Do cross-multiply • Back to the given integral and do the substitution • Take the integral • Express the final answer in term of x

  3. Yah! We are done. • What have you studied in MAT 116? • What is Calculus?

  4. What is “calculus”? • Calculus is the study of how things change. • What do you learn in a calculus class? • Functions, limits, derivatives, integrals, … • Derivatives can used to study applications that involve the rate of change such as population growth, radioactive decay, optimization problems, … • Integrals can be used to calculate the area, equation of a curve, business sales, … • How do algebra and calculus differ? Ex: Solving equations to find a particular value of a variable Ex: How a change in a variable affects another variable?

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