Question

$\int e^{2x}\sin xdx$

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int e^{2x}\sin xdx \\ \mathrm{Considering}\sin x\mathrm{as}\mathrm{first}\mathrm{function}\mathrm{and}{e}^{2x}\mathrm{as}\mathrm{second}\mathrm{function} \\ I=\sin x\frac{e^{2x}}{2}-\int \cos x\frac{e^{2x}}{2}dx \\ \Rightarrow I=\sin x\frac{e^{2x}}{2}-\frac{1}{2}\int \cos x{e}^{2x}dx \\ \Rightarrow I=\frac{\sin x{e}^{2x}}{2}-\frac{1}{2}\left[\cos x\frac{e^{2x}}{2}-\int \left(-\sin x\right)\frac{e^{2x}}{2}dx\right] \\ \Rightarrow I=\frac{\sin x{e}^{2x}}{2}-\frac{\cos x{e}^{2x}}{4}-\frac{1}{2}\int \frac{e^{2x}\sin x}{2}dx \\ I=\frac{e^{2x}\left(2\sin x-\cos x\right)}{4}-\frac{I}{4} \\ \Rightarrow 5I=e^{2x}\left(2\sin x-\cos x\right) \\ \Rightarrow I=\frac{e^{2x}\left(2\sin x-\cos x\right)}{5}+\mathrm{C} \end{gathered}$$