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)

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