Question

Solve : $\int e^x.\sin 3xdx$

Answer 1

$\int e^x\sin 3xdx$

Using Integration by Parts

Considering $u=e^x$ and $v=\sin 3x$

$=e^x(-\frac{1}{3}\cos 3x)-\int e^x(-\frac{1}{3}\cos 3x)dx$

$=e^x(-\frac{1}{3}\cos 3x)+\frac{1}{3}\int e^x\left(\cos 3x\right)dx$

$=e^x(-\frac{1}{3}\cos 3x)+\frac{1}{3}(e^x\frac{1}{3}\sin 3x-\frac{1}{3}\int e^x\sin 3xdx)$

$=-\frac{1}{10}{e}^x(3\cos 3x-\sin 3x)+C$