Question

Evaluate the following definite integral. \int_{0}^{\pi}x\sin xdx

Applications of integrals
ANSWERED
asked 2021-03-09
Evaluate the following definite integral.
\(\displaystyle{\int_{{{0}}}^{{\pi}}}{x}{\sin{{x}}}{\left.{d}{x}\right.}\)

Answers (1)

2021-03-10
\(\displaystyle{\int_{{{0}}}^{{\pi}}}{x}{\sin{{x}}}{\left.{d}{x}\right.}\)
Integrate by parts:
\(\displaystyle\int{f}{g}'={f}{g}-\int{f}'{g}\)
f=x, f'=1
\(\displaystyle{g}'={\sin{{x}}},{g}=-{\cos{{x}}}\)
\(\displaystyle\Rightarrow{{\left[-{x}{\cos{{x}}}\right]}_{{{0}}}^{{\pi}}}-{\int_{{{0}}}^{{\pi}}}-{\cos{{x}}}{\left.{d}{x}\right.}\)
Solve \(\displaystyle{\int_{{{0}}}^{{\pi}}}-{\cos{{x}}}{\left.{d}{x}\right.}\)
Apply linearity:
\(\displaystyle-{\int_{{{0}}}^{{\pi}}}{\cos{{x}}}{\left.{d}{x}\right.}\)
Solve \(\displaystyle{\int_{{{0}}}^{{\pi}}}{\cos{{x}}}{\left.{d}{x}\right.}\)
It is a standard integral:
\(\displaystyle-{\int_{{{0}}}^{{\pi}}}{\cos{{x}}}{\left.{d}{x}\right.}=-{{\left[{\sin{{x}}}\right]}_{{{0}}}^{{\pi}}}\)
Plugging in solved integrals,
\(\displaystyle{{\left[{\sin{{x}}}-{x}{\cos{{x}}}\right]}_{{{0}}}^{{\pi}}}\)
\(\displaystyle={\sin{{\left(\pi\right)}}}-\pi{\cos{{\left(\pi\right)}}}-{\left[{\sin{{\left({0}\right)}}}-{0}{\cos{{\left({0}\right)}}}\right]}\)
\(\displaystyle={0}-\pi{\left(-{1}\right)}-{\left[{0}-{0}\right]}\)
\(\displaystyle=\pi\)
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