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

The integral of cos^5(5x)sin(5x) is -1/30 cos^6(5x)+C

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integral of cos^5(5x)sin(5x)

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

- \frac{1}{30} cos^{6} (5 x) + C

Solution steps

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

Apply u-substitution

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

Apply the Power Rule

: \frac{5}{6} u^{\frac{6}{5}}

= - \frac{1}{25} \cdot \frac{5}{6} u^{\frac{6}{5}}

Substitute back

u = cos^{5} (5 x)

= - \frac{1}{25} \cdot \frac{5}{6} (cos^{5} (5 x))^{\frac{6}{5}}

Simplify

- \frac{1}{25} \cdot \frac{5}{6} (cos^{5} (5 x))^{\frac{6}{5}} : - \frac{1}{30} cos^{6} (5 x)

= - \frac{1}{30} cos^{6} (5 x)

Add a constant to the solution

= - \frac{1}{30} cos^{6} (5 x) + C

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What is the integral of cos^5(5x)sin(5x) ?
The integral of cos^5(5x)sin(5x) is -1/30 cos^6(5x)+C