Integration of x e^2x /(1+2x)^2

 

$$\begin{array}{l}\int \frac{x{\mathrm{e}}^{{\mathbf{2}}^x}}{{\left(\mathbf{1}+\mathbf{2}x\right)}^{\mathbf{2}}}dx\\ \\ ByPart\\ u=x{e}^{2x}dv={\left(1+2x\right)}^{-2}\\ du=(e^{2x}+x{e}^{2x})dxv=\frac{-1}{2\left(1+2x\right)}\\ \\ \frac{-x{e}^{2x}}{2\left(1+2x\right)}+\int \frac{\left(1+2x\right)e^{2x}}{2\left(1+2x\right)}\mathrm{d}x\\ \\ \Rightarrow \frac{-\mathrm{x}{e}^{2x}}{\overline{\left(1+2x\right)}2}+\frac{e^{2x}}{4}+c\end{array}$$