Question

If $\int x\sin xdx$ = −x cos x + α, then α is equal to
(a) sin x + C
(b) cos x + C
(c) C
(d) none of these

Answer 1

(a) sin x + C

$$\begin{gathered} \int \underset{\mathrm{I}}{x}·\underset{\mathrm{II}}{\sin x}dx=-x\cos x+\alpha \\ \Rightarrow x\int \sin xdx-\int \left\{\frac{d}{dx}\left(x\right)\int \sin xdx\right\}dx=-x\cos x+\alpha \\ \Rightarrow x\left(-\cos x\right)-\int 1·\left(-\cos x\right)dx=-x\cos x+\alpha \\ \Rightarrow -x\cos x+\int \cos xdx=-x\cos x+\alpha \\ \Rightarrow -x\cos x+\sin x+C=-\mathrm{x}\cos \mathrm{x}+\mathrm{\alpha } \\ \therefore \alpha =\sin x+C \end{gathered}$$