What is the integral of sin(9x)cos^5(9x) ?

The integral of sin(9x)cos^5(9x) is -1/54 cos^6(9x)+C

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integral of sin(9x)cos^5(9x)

\int sin (9 x) cos^{5} (9 x) d x

- \frac{1}{54} cos^{6} (9 x) + C

Solution steps

\int sin (9 x) cos^{5} (9 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{1}{45} \cdot \frac{5}{6} u^{\frac{6}{5}}

Substitute back

u = cos^{5} (9 x)

= - \frac{1}{45} \cdot \frac{5}{6} (cos^{5} (9 x))^{\frac{6}{5}}

Simplify

- \frac{1}{45} \cdot \frac{5}{6} (cos^{5} (9 x))^{\frac{6}{5}} : - \frac{1}{54} cos^{6} (9 x)

= - \frac{1}{54} cos^{6} (9 x)

Add a constant to the solution

= - \frac{1}{54} cos^{6} (9 x) + C

What is the integral of sin(9x)cos^5(9x) ?
The integral of sin(9x)cos^5(9x) is -1/54 cos^6(9x)+C