I need to do an integral for my quantum mechanics homework but it has been a while since I've taken calc so my math is a bit rusty.

The first integral I can do, it's the second one that I'm having trouble with. I tried by parts and know that I can do it with brute force, but the it involves using by parts multiple times. In physics, we are usually given less involve math problems, so I was just wondering whether there is a more elegant way to do this.