What is the integral of cos^3(5x)sin^{-2}(5x) ?

The integral of cos^3(5x)sin^{-2}(5x) is -1/5 cos^3(5x)cot(5x)-3/5 sin(5x)+1/5 sin^3(5x)+C

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integral of cos^3(5x)sin^{-2}(5x)

\int cos^{3} (5 x) sin^{- 2} (5 x) d x

- \frac{1}{5} cos^{3} (5 x) cot (5 x) - \frac{3}{5} sin (5 x) + \frac{1}{5} sin^{3} (5 x) + C

Solution steps

\int cos^{3} (5 x) sin^{- 2} (5 x) d x

Apply Integration By Parts

: - \frac{1}{5} cos^{3} (5 x) cot (5 x) - \int 3 cos^{2} (5 x) sin (5 x) cot (5 x) d x

= - \frac{1}{5} cos^{3} (5 x) cot (5 x) - \int 3 cos^{2} (5 x) sin (5 x) cot (5 x) d x

\int 3 cos^{2} (5 x) sin (5 x) cot (5 x) d x = \frac{3}{5} sin (5 x) - \frac{1}{5} sin^{3} (5 x)

= - \frac{1}{5} cos^{3} (5 x) cot (5 x) - (\frac{3}{5} sin (5 x) - \frac{1}{5} sin^{3} (5 x))

Simplify

= - \frac{1}{5} cos^{3} (5 x) cot (5 x) - \frac{3}{5} sin (5 x) + \frac{1}{5} sin^{3} (5 x)

Add a constant to the solution

= - \frac{1}{5} cos^{3} (5 x) cot (5 x) - \frac{3}{5} sin (5 x) + \frac{1}{5} sin^{3} (5 x) + C

\int 2 e^{- 3 x} d x

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

sin^{2} (y) d x + cos^{2} (x) d y = 0

x \to - 5 lim (2 x + 3)

\frac{1 + e ^{- π s}}{s ^{2} + 1}

What is the integral of cos^3(5x)sin^{-2}(5x) ?
The integral of cos^3(5x)sin^{-2}(5x) is -1/5 cos^3(5x)cot(5x)-3/5 sin(5x)+1/5 sin^3(5x)+C