What is the integral of cos^8(6x)sin(6x) ?

The integral of cos^8(6x)sin(6x) is -1/54 cos^9(6x)+C

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integral of cos^8(6x)sin(6x)

\int cos^{8} (6 x) sin (6 x) d x

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

Solution steps

\int cos^{8} (6 x) sin (6 x) d x

Apply u-substitution

: \int cos^{8} (u) sin (u) \frac{1}{6} d u

= \int cos^{8} (u) sin (u) \frac{1}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int cos^{8} (u) sin (u) d u

Apply u-substitution

: \int - v^{8} d v

= \frac{1}{6} \cdot \int - v^{8} d v

Take the constant out:

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

= \frac{1}{6} (- \int v^{8} d v)

Apply the Power Rule

: \frac{v ^{9}}{9}

= \frac{1}{6} (- \frac{v ^{9}}{9})

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of cos^8(6x)sin(6x) ?
The integral of cos^8(6x)sin(6x) is -1/54 cos^9(6x)+C