Integrate (xe^(2x))/(1+2x)^2

autodidude

1. The problem statement, all variables and given/known data
Integrate [tex]\frac{xe^{2x}}{(1+2x)^2}[/tex] with respect to x

Didn't get anywhere with integration by parts or substitution using u=xe^(2x)
A push in the right direction would be much appreciated.

voko

Try v = 1 + 2x.

autodidude

As a second substitution?

voko

No, just start with it.

autodidude

Ok, I now have the following:

[tex]\frac{1}{4} \int \frac{(u-1)e^{(u-1)}{u^2}[/tex]

Karnage1993

Allow me to fix that for you:

##\displaystyle \frac{1}{4} \int \frac{(u-1)e^{(u-1)}}{u^2} \ du##

voko

where is du?

ehild

Integrate by parts

∫uv'dx=uv-∫u'vdx,

using u=xe^2x and v'=1/(1+2x)².

ehild

Karnage1993

Parts requires u,v to be continuous.

voko

Now that we have reinstated du, observe that e^(u - 1) = (e^u)/e; the 1/e constant goes outside, and what's inside can be simplified into ((e^u)/u - (e^u)/u^2).