8.2 Integration by Parts

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, arcsin ax, or arctan ax and let dv = xn dx 3. For integrals of the form or Let u = sin bx or cos bx and let dv = eax dx

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

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.

Evaluate u dv u dv u dv u dv 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.

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

Since x2 integrates easier than ln x, let u = ln x and dv = x2 u = ln x dv = x2 dx

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

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