Question

$\int e^{x}sinx$ $dx$

Answer 1

We have,

$\int e^x\sin xdx$

Using product rule and we get,

$\int I.IIdx=I\int IIdx-\int (\frac{dI}{dx}\int IIdx)dx$

And we get,

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

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

$I=\sin x{e}^x-[\cos x\int e^{x}dx-\int \frac{d}{dx}\cos x\int e^{x}dx]$

$I=\sin x{e}^x-[e^x\cos x-\int e^x(-\sin x)]$

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

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

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

$2I=e^x\sin x-e^x\cos x$

$I=\frac{1}{2}{e}^x(\sin x-\cos x)$

Hence, this is the answer.