Integration by Parts: Definite Integrals Examples

Find

First we integrate the corresponding indefinite integral using integration by parts. We find that

Since

xe^x – e^x + C

is the family of all antiderivatives of xe^x, we can get one particular antiderivative by taking C = 0. Then we can use the resulting antiderivative

xe^x – e^x

in the FTC.

Find

We take

u = x

v' = e^x

u' = 1

v = e^x.

Then we apply the formula, keeping the limits of integration:

Finally we evaluate each term from 0 to 1:

It's reassuring that we got the same answer we got with Way 1.