What is the integral of cos^3(3θ)sin^{-2}(3θ) ?

The integral of cos^3(3θ)sin^{-2}(3θ) is -1/3 cos^3(3θ)cot(3θ)-sin(3θ)+1/3 sin^3(3θ)+C

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integral of cos^3(3θ)sin^{-2}(3θ)

\int cos^{3} (3 θ) sin^{- 2} (3 θ) d θ

- \frac{1}{3} cos^{3} (3 θ) cot (3 θ) - sin (3 θ) + \frac{1}{3} sin^{3} (3 θ) + C

Solution steps

\int cos^{3} (3 θ) sin^{- 2} (3 θ) d θ

Apply Integration By Parts

= - \frac{1}{3} cos^{3} (3 θ) cot (3 θ) - \int 3 cos^{2} (3 θ) sin (3 θ) cot (3 θ) d θ

\int 3 cos^{2} (3 θ) sin (3 θ) cot (3 θ) d θ = sin (3 θ) - \frac{1}{3} sin^{3} (3 θ)

= - \frac{1}{3} cos^{3} (3 θ) cot (3 θ) - (sin (3 θ) - \frac{1}{3} sin^{3} (3 θ))

Simplify

= - \frac{1}{3} cos^{3} (3 θ) cot (3 θ) - sin (3 θ) + \frac{1}{3} sin^{3} (3 θ)

Add a constant to the solution

= - \frac{1}{3} cos^{3} (3 θ) cot (3 θ) - sin (3 θ) + \frac{1}{3} sin^{3} (3 θ) + C

What is the integral of cos^3(3θ)sin^{-2}(3θ) ?
The integral of cos^3(3θ)sin^{-2}(3θ) is -1/3 cos^3(3θ)cot(3θ)-sin(3θ)+1/3 sin^3(3θ)+C