Hard integration problem (calc 2)

[Azi]

I've been going at this problem for at least 3 hours

Just need to find the integral for:

INTEGRAL(x^3e^x^2) / (x^2 + 1)^2 dxI've tried all kinds of different methods, filling up at 4 pages in my notebook! I think this is the closest I've gotten

First I broke up the numerator so that it's: x^2xe^x^2
Attempting to solve with u-substitution

u = x^2 + 1
x^2 = u - 1
du = 2xdx
1/2du = xdx

1/2INTEGRAL((u-1)e^(u-1)) / (u^2) du

And I don't know what to do from here, I don't feel like I'm on the right track. If anyone has a tip it would be much appreciated!

 

[Bohrok]

Try letting u = x²; that way you don't have to work with (u - 1)e^{u - 1} and you get e^uu.

 

[Azi]

Hmm yeah that probably is the best choice, I tried it earlier but am still hitting a wall

u = x^2
du = 2xdx
1/2du = xdx

1/2INTEGRAL (Ue^U) / (U+1)^2 dU

Can that be integrated? I've always been so bad at knowing when I can do something and when I can't, it kills me on tests.

 

[Bohrok]

Looks like your only option now is integration by parts, seems like no other kinds of substitutions would help. You only have a few things to try with u, e^u, and 1/(u + 1)², shouldn't be too hard.

 

[Azi]

Thanks! I think I may have gotten it, not 100% sure though, got class soon so i'll find out then. Thanks again