Integration by Parts: linear times exp(linear): Find

We shall try Integration by Parts, with

and . We hope that the new integral, , will be easier than the original one. The method also depends on du being simpler than u. In particular, and we see that we can use the integral of the exponential:

.

Now we substitute these parts into the original problem:

, or

and we see that the new integral is easier than the original since it involves just the exponential, not x and the exponential.

Carrying out the last integral and simplifying, we find

, or