Question

$\int x{e}^{2x}dx$

Answer 1

$$\begin{gathered} \int x{e}^{2x}dx \\ \mathrm{Taking}x\mathrm{as}\mathrm{the}\mathrm{first}\mathrm{function}\mathrm{and}{e}^{2x}\mathrm{as}\mathrm{the}\mathrm{second}\mathrm{function}. \\ =x\int e^{2x}dx-\int \left\{\frac{d}{dx}\left(x\right)\int e^{2x}dx\right\}dx \\ =\frac{x{e}^{2x}}{2}-\int \left(\frac{e^{2x}}{2}\right)dx \\ =\frac{x}{2}{e}^{2x}-\frac{e^{2x}}{4}+C \\ =e^{2x}\left(\frac{x}{2}-\frac{1}{4}\right)+C \end{gathered}$$