Struggling with Integrating (xe^(2x))/(1+2x)^2? Get a Helpful Tip Here!

[autodidude]

Homework Statement

Integrate $$\frac{xe^{2x}}{(1+2x)^2}$$ 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:

$$\frac{1}{4} \int \frac{(u-1)e^{(u-1)}{u^2}$$

 

[Karnage1993]

Ok, I now have the following:

$$\frac{1}{4} \int \frac{(u-1)e^{(u-1)}{u^2}$$

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]

Homework Statement

Integrate $$\frac{xe^{2x}}{(1+2x)^2}$$ 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.

Integrate by parts

∫uv'dx=uv-∫u'vdx,

using u=xe^2x and v'=1/(1+2x)².

ehild

 

[Karnage1993]

Integrate by parts

∫uv'dx=uv-∫u'vdx,

using u=xe^2x and v'=1/(1+2x)².

ehild

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).