allery$\int x\cos x\,dx$ equals
A$x\sin x+\cos x+C$
B$x\sin x-\cos x+C$
C$x\cos x+\sin x+C$
D$\dfrac{x^2}{2}\sin x+C$
Answer & Solution
Correct answer: A. $x\sin x+\cos x+C$
1. Take $u=x$ (first), $dv=\cos x\,dx$, so $v=\sin x$.
2. By parts: $x\sin x-\int \sin x\,dx$.
3. $=x\sin x-(-\cos x)=x\sin x+\cos x+C$.
4. Option B drops the sign correction on $\cos x$.
_Source: syllabus Mathematics Ch 7 "Integrals", p.324_
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