Question

Integrate :$\int \:e^x\cos \left(x\right)dx$

Answer 1

$\int \:e^x\cos \left(x\right)dx=\frac{e^x\sin \left(x\right)}{2}+\frac{e^x\cos \left(x\right)}{2}+C$

$\mathrm{Apply\:Integration\:By\:Parts:}\:u=e^x,\:v'=\cos \left(x\right)$

$u'=\frac{d}{dx}\left(e^x\right)=e^x$

$v=\int \cos \left(x\right)dx=\sin \left(x\right)$

$=e^x\sin \left(x\right)-\int \:e^x\sin \left(x\right)dx$

$\mathrm{Apply\:Integration\:By\:Parts:}\:u=e^x,\:v'=\sin \left(x\right)$

$=e^x\sin \left(x\right)-\left(-e^x\cos \left(x\right)-\int \:-e^x\cos \left(x\right)dx\right)$

$\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx$

$=e^x\sin \left(x\right)-\left(-e^x\cos \left(x\right)-\left(-\int \:e^x\cos \left(x\right)dx\right)\right)$

$\mathrm{Therefore}$

$\int \:e^x\cos \left(x\right)dx=e^x\sin \left(x\right)-\left(-e^x\cos \left(x\right)-\left(-\int \:e^x\cos \left(x\right)dx\right)\right)$

$\mathrm{Isolate}\:\int \:e^x\cos \left(x\right)dx$

$=\frac{e^x\sin \left(x\right)}{2}+\frac{e^x\cos \left(x\right)}{2}$

$Add\:a\:constant\:to\:the\:solution$

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

Answer 2

Explanation:

Hope it helps You dear....!!