intergral of e^x / (1-2e^x)^3 dx help

e^x / (1-2e^x)^3 dx

i replaced (1-2e^x) with a random letter in my case i used "p"
i ended up with the answer: 1 / 4(1-2e^x)² + C
however it doesnt look right to me, can anyone confirm if i am right or wrong?

Re: intergral of e^x / (1-2e^x)^3 dx help

Quote:

Originally Posted by bobmarly12345

e^x / (1-2e^x)^3 dx

i replaced (1-2e^x) with a random letter in my case i used "p"
i ended up with the answer: 1 / 4(1-2e^x)² + C
however it doesnt look right to me, can anyone confirm if i am right or wrong?

First note that $\displaystyle \begin{align*} \int{\frac{e^x}{\left(1 - 2e^x\right)^3}\,dx} = -\frac{1}{2}\int{\frac{-2e^x}{\left(1 - 2e^x\right)^3}\,dx} \end{align*}$ .

Now make the substitution $\displaystyle \begin{align*} u = 1 - 2e^x \implies du = -2e^x\,dx \end{align*}$ and the integral becomes

$\displaystyle \begin{align*} -\frac{1}{2}\int{\frac{1}{u^3}\,du} = -\frac{1}{2}\int{u^{-3}\,du} \end{align*}$ .

I'm sure you can go from here :)

Edit: On closer inspection of your original post, you got the answer right :)

Re: intergral of e^x / (1-2e^x)^3 dx help

It is correct. You can check by seeing whether you get back to the question when you differentiate your answer.