how do you integrate e^x - xe^x dx (limits from 0 to 1) without having to use integration by parts?

Question

anonymous · Answer

why the handicap?

anonymous · Answer

we haven't studies ibp yet, but this problem is in our homework

anonymous · Answer

it is not a u sub
it is not a trig sub
what method can you use?

anonymous · Answer

i wouldn't even know, i'm barely starting calc 2 so not familiar with the techniques.  I'm just going to bring this to the professors attention.

anonymous · Answer

maybe some gimmick i don't see like calling it $I$ and getting some equation in $I$ that you can solve for
not seeing it though, sorry

anonymous · Answer

it's ok, i appreciate your help

kainui · Answer

Well the left one is super easy, no problem there. The real thing you're trying to solve is to within being multiplied by a constant (-1 in this case): $$\LARGE \int\limits xe^x dx$$

But notice, what is the derivative of xe^x? $$\LARGE (xe^x)'= e^x+xe^x$$ Hey, now let's just integrate both sides.  $$\LARGE xe^x=\int\limits  e^xdx+\int\limits xe^xdx$$ We can just solve for the integral we want now, since the other integral is mega easy. Ok, so all we did was use the product rule. Fancy.

kainui · Answer

(secretly this is actually integration by parts... shhhhh)

anonymous · Answer

yes it sure is isn't it?

kainui · Answer

Well it's sort of just the product rule which you should already know, plus it's exploiting the fact that this function's derivative is really itself plus something else. You can't exactly do this trick normally so I consider it safe to play, idk.

If there's a cleverer trick out there for doing this I'd love to hear it. The only other alternative I can think of is if you somehow learned power series already.