What is the integral of θ^2cos(2θ) ?

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

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integral of θ^2cos(2θ)

\int θ^{2} cos (2 θ) d θ

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

Solution steps

\int θ^{2} cos (2 θ) d θ

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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