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  4. ∫ ex( cos x - sin x)dx is equal to

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Q. $\int e^{x}\left(\cos x -\sin x\right)dx $ is equal to

COMEDKCOMEDK 2014Integrals

Solution:

Let $I = \int e^{x}\left(\cos x -\sin x\right)dx$
$ = \int e^{x} \cos x dx -\int e^{x} \sin x dx $
$= \int e^{x} \cos x dx -\left[ e^{x}\left(- \cos x\right) -\int e^{x} \left(- \cos x\right)dx \right] $
$=\int e^{x} \cos x dx +e^{x} \cos x -\int e^{x} \cos x dx$
$ = e^{x} \cos x +C $