What is the integral of cos^3(7x)sin^{-2}(7x) ?

The integral of cos^3(7x)sin^{-2}(7x) is -1/7 cos^3(7x)cot(7x)-3/7 sin(7x)+1/7 sin^3(7x)+C

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integral of cos^3(7x)sin^{-2}(7x)

\int cos^{3} (7 x) sin^{- 2} (7 x) d x

- \frac{1}{7} cos^{3} (7 x) cot (7 x) - \frac{3}{7} sin (7 x) + \frac{1}{7} sin^{3} (7 x) + C

Solution steps

\int cos^{3} (7 x) sin^{- 2} (7 x) d x

Apply Integration By Parts

= - \frac{1}{7} cos^{3} (7 x) cot (7 x) - \int 3 cos^{2} (7 x) sin (7 x) cot (7 x) d x

\int 3 cos^{2} (7 x) sin (7 x) cot (7 x) d x = \frac{3}{7} sin (7 x) - \frac{1}{7} sin^{3} (7 x)

= - \frac{1}{7} cos^{3} (7 x) cot (7 x) - (\frac{3}{7} sin (7 x) - \frac{1}{7} sin^{3} (7 x))

Simplify

= - \frac{1}{7} cos^{3} (7 x) cot (7 x) - \frac{3}{7} sin (7 x) + \frac{1}{7} sin^{3} (7 x)

Add a constant to the solution

= - \frac{1}{7} cos^{3} (7 x) cot (7 x) - \frac{3}{7} sin (7 x) + \frac{1}{7} sin^{3} (7 x) + C

What is the integral of cos^3(7x)sin^{-2}(7x) ?
The integral of cos^3(7x)sin^{-2}(7x) is -1/7 cos^3(7x)cot(7x)-3/7 sin(7x)+1/7 sin^3(7x)+C