Integration: product containing square root. How can I integrate the following expression? 

I have tried using u-substitution, but I am having problems with integrating the entire expression. So far, I have the following: 

Any help on this is highly appreciated!

You have it. $u=1+e^{-kt}$ and $du=-ke^{-kt}dt$, so $dt = -\frac{du}{k}$ where $k$ is just a constant. Rewrite this and you have $\int \sqrt{u}\frac{du}{-k}$, which I bet you know how to integrate!
