Do you have to use parts?. I think a sub is easier.
Make the subs and we get:
Now, let's use tabular integration:
This works well when we have a something such as e or sin that repeats as we take derivatives. It's based off parts, though. So, I reckon we are using parts.
Add up the signed products diagonally and alternate signs.
Someone will come by saying, "this way is easier". Oh well. I wanted to show you tabular integration in the event you have not seen it. The more one knows the better.