Question

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

• A. $e^{x} \cos x + C$
• B. $e^{x} \sin x + C$
• C. $- e^{x} \cos x + C$
• D. $- e^{x} \sin x + C$

Answer 1

Answer: (A)

$e^{x} \cos x + C$

Answer 2

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