Integrals using Integration by Parts Click here for a discussion of strategy.

In a Nut Shell: Integration by parts is very useful for integrals of the form I = ∫ u dv.

The basic relation used for Integration by Parts is:

I = ∫ u dv = uv - ∫ v du ← This is the basic relation

Example: ∫( ln x /x² )dx

let u = ln x, dv = (1/ x²) dx ← Standard form for integral

Here ∫ (1/ x²) dx is a standard integral of the form ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c

So du = (1/x)dx and v = ˗ (1/x)

Then I becomes -(1/x) ln x + ∫(1/x²⁾ du

with the standard form again ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c

Result: I = ˗ (1/x) ln x - 1/x + c

c is the constant of integration

Method: Multiple Use of Integration by Parts

I = ∫ u dv = uv ˗ ∫ v du

Example: I = ∫ x² e^2x dx

let u = x², dv = e^2x dx ← Standard integral

∫ e^ax dx = (1/a) e^ax + c

du = 2x dx v = ½ e^2x

Then I becomes ½ x²e^2x - ∫x e^2x dx and integrate by parts again,

let u = x, dv = e^2x dx ← Standard integral ∫ e^ax dx

du = dx v = ½ e^2x

Result: I = ½ x²e^2x ˗ ½ x e^2x + ¼ e^2x + c

c is the constant of integration

Return to Notes for Calculus 2

Copyright © 2017 Richard C. Coddington

All rights reserved.