What is the integral of θsin^2(θ) ?

The integral of θsin^2(θ) is 1/2 θ(θ-1/2 sin(2θ))-1/2 ((θ^2)/2+1/4 cos(2θ))+C

integral of θsin^2(θ)

\int θ sin^{2} (θ) d θ

\frac{1}{2} θ (θ - \frac{1}{2} sin (2 θ)) - \frac{1}{2} (\frac{θ ^{2}}{2} + \frac{1}{4} cos (2 θ)) + C

Solution steps

\int θ sin^{2} (θ) d θ

Apply Integration By Parts

= \frac{1}{2} θ (θ - \frac{1}{2} sin (2 θ)) - \int \frac{1}{2} (θ - \frac{1}{2} sin (2 θ)) d θ

\int \frac{1}{2} (θ - \frac{1}{2} sin (2 θ)) d θ = \frac{1}{2} (\frac{θ ^{2}}{2} + \frac{1}{4} cos (2 θ))

= \frac{1}{2} θ (θ - \frac{1}{2} sin (2 θ)) - \frac{1}{2} (\frac{θ ^{2}}{2} + \frac{1}{4} cos (2 θ))

Add a constant to the solution

= \frac{1}{2} θ (θ - \frac{1}{2} sin (2 θ)) - \frac{1}{2} (\frac{θ ^{2}}{2} + \frac{1}{4} cos (2 θ)) + C

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What is the integral of θsin^2(θ) ?
The integral of θsin^2(θ) is 1/2 θ(θ-1/2 sin(2θ))-1/2 ((θ^2)/2+1/4 cos(2θ))+C