Question

Integrate:

$\int e^x\sin x.dx$

Answer 1

$I=\int e^x\sin xdx=e^x\int \sin xdx+\int [\frac{d}{dx}{e}^x.\int \sin xdx]dx=-e^x\cos x-\int e^x\cos xdx=-e^x\cos x-[e^x\int \cos xdx+\int [\frac{d}{dx}{e}^x.\int \cos xdx]dx]=-e^x\cos x-e^x\sin x-\int e^x\sin xdxI=-e^x\cos x-e^x\sin x-I2I=-e^x(\cos x+\sin x)I=-\frac{e^x}{2}(\cos x+\sin x)$