Solving Integral Problem: \int\frac{x*e^x}{(x+1)^2}dx

[clackulus]

The problem is given as
$$\int\frac{x*e^x}{(x+1)^2}dx$$

I did u substitution with u=(x+1) and du=dx

which gives me $$\int\frac{(u-1)*e^{u-1}}{u^2}$$

simplifies to $$\int\frac{u*e^{u-1}-e^{u-1}}{u^2}$$

Then I separated it into two integrals

$$\int\frac{e^{u-1}}{u}-\int\frac{e^{u-1}}{u^2}$$


Now I'm stuck. I tried doing these separate integrals by parts, but it doesn't seem to be working for me. Am I going in the complete wrong direction with this? Any help would be appreciated.

 

[gabbagabbahey]

Everything looks good to me so far, what do you get when you integrate the second term by parts once?

 

[clackulus]

Oh wow. I got caught up on the fact that I couldn't do the first term and didn't realize that I don't have to! The integral of the second term takes care of that for me. Thank you.