Question

Evaluate: $I=\int e^{2x}\sin xdx$

Answer 1

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

$I=\frac{e^{2x}}{2}\sin x+\int \frac{e^{2x}}{2}\cos xdx$

$I=\frac{e^{2x}}{2}\sin x+\frac{e^{2x}}{4}\cos x-\int \frac{e^{2x}}{4}\sin xdx$

$I=\frac{e^{2x}}{2}\sin x+\frac{e^{2x}}{4}\cos x-\frac{I}{4}+c$

$\frac{5I}{4}=\frac{e^{2x}\sin x}{2}+\frac{e^{2x}\cos x}{4}+c$

$I=\frac{2e^{2x}\sin x}{5}+\frac{e^{2x}\cos x}{5}+c$