How do you solve this calculus problem?

Question

Can you help me solve this calculus problem?
How do you solve this calculus problem?
Can you at least tell me u, du, v, dv?
What is an antiderivative of interval (xe^x)/[(x+1)^2]?
Please include the steps to do the problem. Thank you.

Answer 1

∫xe^x/(x+1)² dx

First, break this into two integrals:

∫(x+1)e^x/(x+1)² dx - ∫e^x/(x+1)² dx

Simplify:

∫e^x/(x+1) dx - ∫e^x/(x+1)² dx

Integrate the first one by parts -- let u = 1/(x+1), v = e^x, du = -1/(x+1)² dx, dv = e^x dx. So we have:

e^x/(x+1) - ∫-e^x/(x+1)² dx - ∫e^x/(x+1)² dx
e^x/(x+1) + ∫e^x/(x+1)² dx - ∫e^x/(x+1)² dx
e^x/(x+1) + ∫0 dx
e^x/(x+1) + C

And we are done.