Question

Evaluate $\int e^x\sin xdx$

• A. $-e^x(\cos x-\sin x)$
• B. $\frac{-e^x}{2}(\cos x-\sin x)$
• C. $\frac{-e^x}{2}(\cos x+\sin x)$
• D. $-e^x(\cos x+\sin x)$

Answer 1

The correct option is B $\frac{-e^x}{2}(\cos x-\sin x)$

Let $I=\int e^x\sin xdx$
Using Integration by parts method we have,
$\int e^x\sin xdx=e^x\int \sin xdx-\int (\frac{d{e}^x}{dx}\int \sin xdx)dx$
$\Rightarrow -e^x\cos x+\int e^x\cos xdx$
$\Rightarrow -e^x\cos x+e^x\int \cos xdx-\int (\frac{d{e}^x}{dx}\int \cos xdx)dx$
$\Rightarrow -e^x\cos x+e^x\sin x-\int e^x\sin xdx$
$\Rightarrow I=-e^x\cos x+e^x\sin x-\int e^x\sin xdx$
$\Rightarrow I=-e^x\cos x+e^x\sin x-I$
$\Rightarrow 2I=-e^x(\cos x-\sin x)$
$\Rightarrow I=\frac{-e^x}{2}(\cos x-\sin x)$