Section 8.2 Integration by Parts

1. Section 8.2 Integration by Parts

2. Summary of Common Integrals Using Integration by Parts 1. For integrals of the form Let u = xn and let dv = eax dx, sin ax dx, cos ax dx 2. For integrals of the form Let u = lnx, arcsinax, or arctanax and let dv = xn dx 3. For integrals of the form or Let u = sin bx or cos bx and let dv= eax dx

3. Integration by Parts If u and v are functions of x and have continuous derivatives, then

4. Guidelines for Integration by Parts • Try letting dv be the most complicated • portion of the integrand that fits a basic • integration formula. Then u will be the • remaining factor(s) of the integrand. • Try letting u be the portion of the • integrand whose derivative is a simpler • function than u. Then dv will be the • remaining factor(s) of the integrand.

5. Example 1

6. Evaluate To apply integration by parts, we want to write the integral in the form There are several ways to do this. u dv u dv u dv u dv Following our guidelines, we choose the first option because the derivative of u = x is the simplest and dv = ex dx is the most complicated.

7. u = x v = ex du = dx dv = ex dx u dv

8. Example 2

9. Since x2 integrates easier than lnx, let u = lnx and dv = x2 u = lnx dv = x2dx

10. Example 3

11. Repeated application of integration by parts u = x2 v = -cos x du = 2x dx dv = sin x dx Apply integration by parts again. u = 2x du = 2 dx dv = cos x dx v = sin x

12. Example 4

13. Repeated application of integration by parts Neither of these choices for u differentiate down to nothing, so we can let u = exor sin x. v = ex Let’s let u = sin x du = cos x dx dv = ex dx u = cos x v = ex du = -sin x dx dv = ex dx