Question

Integrate with respect to $x$:
$e^x\sin x$

Answer 1

Consider the following integral.

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

Using a ilate rule in a given integral.

$I=e^x\underset{}{\overset{}{\int }}\sin xdx-\underset{}{\overset{}{\int }}\left(\frac{d\left(e^x\right)}{dx}\underset{}{\overset{}{\int }}\left(\sin x\right)\right)dx$

$I=-e^x\cos x+\underset{}{\overset{}{\int }}{e}^x\cos xdx$

$I=-e^x\cos x+e^x\sin x-\underset{}{\overset{}{\int }}{e}^x\sin xdx$

$I=-e^x\cos x+e^x\sin x-I$

$2I=e^x(\sin x-\cos x)+C$

$I=\frac{e^x}{2}(\sin x-\cos x)+C$

Hence, this is the required answer.