Question

Solve the integral of (x*e^2x)/(1+2x)^2 by first using u substitution and then use integration by parts.


I am very confused on how to do u-substitution with this because no mattter what I let u be equal to the du never comes out to be anything in the equation?

The correct answer is suppose to be (e^2x)/(4*(1+2x)) + C but I don't know how they got that?

Answer 1

Sorry about that, it just says to solve it and the problem didn't say that you need u sub for it.

However I still don't know what to let u and dv be?

Answer 2

Cute. Let

u = xe^(2x)
du = e^(2x)(1+2x)
dv = 1/(1+2x)^2
v = (-1/2)/(1+2x)

∫u dv = uv - ∫v du
= -(xe^(2x))/(2(1+2x)) - ∫(-1/2)e^(2x) dx
= -(xe^(2x))/(2(1+2x)) + e^(2x)/4
= e^(2x)/(4(1+2x)) + C

Answer 3

Thank you so much for your help!