Integrate the function x sin x - Vidyadip

Question

Integrate the function x sin x

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Answer

$\int {x\sin x} dx$

$= x\int {\sin x} dx - \int {\left( {\frac{d}{{dx}}x\int {\sin xdx} } \right)dx} $

[Applying product rule]

$ = x\left( { - \cos x} \right) - \int {1\left( { - \cos x} \right)dx} $

$ = -x\cos x - \int { - \cos xdx} $

$= - x\cos x + \int {\cos xdx} $

= -x cos x + sin x + c

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