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

The integral of (sin(3x))^5cos(3x) is 1/18 sin^6(3x)+C

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

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

\frac{1}{18} sin^{6} (3 x) + C

Solution steps

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

Simplify

= \int sin^{5} (3 x) cos (3 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}{15} \cdot \frac{5}{6} u^{\frac{6}{5}}

Substitute back

u = sin^{5} (3 x)

= \frac{1}{15} \cdot \frac{5}{6} (sin^{5} (3 x))^{\frac{6}{5}}

Simplify

\frac{1}{15} \cdot \frac{5}{6} (sin^{5} (3 x))^{\frac{6}{5}} : \frac{1}{18} sin^{6} (3 x)

= \frac{1}{18} sin^{6} (3 x)

Add a constant to the solution

= \frac{1}{18} sin^{6} (3 x) + C

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What is the integral of (sin(3x))^5cos(3x) ?
The integral of (sin(3x))^5cos(3x) is 1/18 sin^6(3x)+C