What is the integral of (1-cos(5x))^3sin(5x) ?

The integral of (1-cos(5x))^3sin(5x) is -1/5 cos(5x)+3/10 cos^2(5x)-1/5 cos^3(5x)+1/20 cos^4(5x)+C

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integral of (1-cos(5x))^3sin(5x)

\int (1 - cos (5 x))^{3} sin (5 x) d x

- \frac{1}{5} cos (5 x) + \frac{3}{10} cos^{2} (5 x) - \frac{1}{5} cos^{3} (5 x) + \frac{1}{20} cos^{4} (5 x) + C

Solution steps

\int (1 - cos (5 x))^{3} sin (5 x) d x

Expand

(1 - cos (5 x))^{3} sin (5 x) : sin (5 x) - 3 sin (5 x) cos (5 x) + 3 cos^{2} (5 x) sin (5 x) - cos^{3} (5 x) sin (5 x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int sin (5 x) d x = - \frac{1}{5} cos (5 x)

\int 3 sin (5 x) cos (5 x) d x = - \frac{3}{10} cos^{2} (5 x)

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

\int cos^{3} (5 x) sin (5 x) d x = - \frac{1}{20} cos^{4} (5 x)

= - \frac{1}{5} cos (5 x) - (- \frac{3}{10} cos^{2} (5 x)) - \frac{1}{5} cos^{3} (5 x) - (- \frac{1}{20} cos^{4} (5 x))

Simplify

= - \frac{1}{5} cos (5 x) + \frac{3}{10} cos^{2} (5 x) - \frac{1}{5} cos^{3} (5 x) + \frac{1}{20} cos^{4} (5 x)

Add a constant to the solution

= - \frac{1}{5} cos (5 x) + \frac{3}{10} cos^{2} (5 x) - \frac{1}{5} cos^{3} (5 x) + \frac{1}{20} cos^{4} (5 x) + C

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

What is the integral of (1-cos(5x))^3sin(5x) ?
The integral of (1-cos(5x))^3sin(5x) is -1/5 cos(5x)+3/10 cos^2(5x)-1/5 cos^3(5x)+1/20 cos^4(5x)+C