Goals : Compute indefinite integrals by the method of substitution.

AP Calculus BC – Chapter 6Differential Equations and Mathematical Modeling 6.2: Antidifferentiation by Substitution- Day 1 Goals: Compute indefinite integrals by the method of substitution. Compute definite integrals by the method of substitution. Solve a differential equation of the form dy/dx = f(x, y) in which the variables are separable.

Properties of Indefinite Integrals: ∫kf(x)dx = k ∫f(x)dx, any constant k. ∫(f(x)±g(x))dx = ∫f(x)dx ± ∫ g(x)dx. Power Formulas: ∫undu = + C, n ≠ 1. ∫u-1du = ∫1/u du = ln|u| + C.

Exploration:

Trigonometric Formulas: ∫cosu du = sinu + C. ∫sinu du = -cosu + C. ∫sec²u du = tanu + C. ∫csc²u du = -cotu + C. ∫secu tanu du = secu + C. ∫cscu cotu du = -cscu + C.

Exponential and Logarithmic Formulas: ∫eu du = eu + C. ∫au du = au/lna + C. ∫lnu du = ulnu – u + C. ∫logau du = ∫lnu/lna du = +C.

An example: How could we evaluate ∫3x²(1 + x³)7dx?

Let’s generalize: Suppose we have functions F and G with corresponding derivatives f and g. The Chain Rule gives: (F(G(x)))’ = F’(G(x))G’(x) = f(G(x))g(x). Take the indefinite integral of LHS & RHS: ∫(F(G(x)))’ dx = ∫f(G(x))g(x)dx. But, ∫(F(G(x)))’dx = F(G(x)) + C. So, ∫f(G(x))g(x)dx = F(G(x)) + C.

Assignment: • HW 6.2A: Read lesson 6.2 through example 7 on page 319 and do the following exercises: #1-11 (odds), 15, 17, 18, 21, 27, 30.

Goals : Compute indefinite integrals by the method of substitution.