What is the integral of cos^3(4x)sin^{-2}(4x) ?

The integral of cos^3(4x)sin^{-2}(4x) is -1/4 cos^3(4x)cot(4x)-3/4 sin(4x)+1/4 sin^3(4x)+C

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integral of cos^3(4x)sin^{-2}(4x)

\int cos^{3} (4 x) sin^{- 2} (4 x) d x

- \frac{1}{4} cos^{3} (4 x) cot (4 x) - \frac{3}{4} sin (4 x) + \frac{1}{4} sin^{3} (4 x) + C

Solution steps

\int cos^{3} (4 x) sin^{- 2} (4 x) d x

Apply Integration By Parts

= - \frac{1}{4} cos^{3} (4 x) cot (4 x) - \int 3 cos^{2} (4 x) sin (4 x) cot (4 x) d x

\int 3 cos^{2} (4 x) sin (4 x) cot (4 x) d x = \frac{3}{4} sin (4 x) - \frac{1}{4} sin^{3} (4 x)

= - \frac{1}{4} cos^{3} (4 x) cot (4 x) - (\frac{3}{4} sin (4 x) - \frac{1}{4} sin^{3} (4 x))

Simplify

= - \frac{1}{4} cos^{3} (4 x) cot (4 x) - \frac{3}{4} sin (4 x) + \frac{1}{4} sin^{3} (4 x)

Add a constant to the solution

= - \frac{1}{4} cos^{3} (4 x) cot (4 x) - \frac{3}{4} sin (4 x) + \frac{1}{4} sin^{3} (4 x) + C

What is the integral of cos^3(4x)sin^{-2}(4x) ?
The integral of cos^3(4x)sin^{-2}(4x) is -1/4 cos^3(4x)cot(4x)-3/4 sin(4x)+1/4 sin^3(4x)+C