Solving integration by parts $\int\frac{xe^{2x}}{(1+2x)^2}\,dx$ $\displaystyle\int\frac{xe^{2x}}{(1+2x)^2}\,dx$
With this particular problem. my approach is to to rewrite the integral as  $$\int xe^{2x}\frac{1}{(1+2x)^2}\,dx$$ and then pick a $u$ and a $dv$ and take it from there. The only issue I'm running into is that $xe^{2x}$ appears to me as two functions instead of one. What is a suggestion for this?

I would suggest letting $u=xe^{2x}$, since it must be integrated by parts (whereas the derivative is the product rule), and $dv=\frac{1}{(1+2x)^2}dx$.
You can allow a composite of two functions be equal to $u$. That is perfectly valid and often required.
