What is the integral of θ^2sin(2θ) ?

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

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integral of θ^2sin(2θ)

\int θ^{2} sin (2 θ) d θ

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

Solution steps

\int θ^{2} sin (2 θ) d θ

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int e^{- s t} t^{3} d t

\frac{d}{d x} ((sin (x)) d x)

x \to 2 lim (\frac{x - 1}{x - 2})

x \to 0 + lim (cos^{\frac{1}{x}} (x))

\frac{\partial}{\partial x} (y^{5} - 3 x y)

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