Practice Integration by Parts in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
An integration technique based on the product rule: \int u\,dv = uv - \int v\,du. Used when the integrand is a product of two functions.
The product rule for derivatives says (uv)' = u'v + uv'. Rearranging and integrating gives integration by parts. The idea is to trade your original integral for a (hopefully easier) one. You're transferring the derivative from one factor to the other.
Example 1
easyFind \displaystyle\int x e^x\,dx.
Example 2
hardFind \displaystyle\int e^x \sin x\,dx.
Example 3
mediumEvaluate \displaystyle\int x^2 e^x \, dx using integration by parts twice.
Example 4
easyFind \displaystyle\int x\cos x\,dx.
Example 5
mediumFind \displaystyle\int \ln x\,dx.