Question:
Mark the correct alternative in each of the following:
If $\int x \sin x d x=-x \cos x+\alpha$, then $\alpha$ is equal to
A. $\sin x+C$
B. $\cos x+C$
C. $\mathrm{C}$
D. none of these
Solution:
using integration by parts
$I=\int x \sin x d \square$
$\left.=x \int \sin x d x-\int \frac{d x}{d x}(x) \int \sin x\right)$
$I=x \cos x+\int \cos x d x$
$\left(\because \int \sin x=-\cos x\right)$
$=x \cos x+\sin x+c$