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

The integral of cos^6(8x)sin(8x) is -1/56 cos^7(8x)+C

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

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

- \frac{1}{56} cos^{7} (8 x) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int - v^{6} d v

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{v ^{7}}{7}

= \frac{1}{8} (- \frac{v ^{7}}{7})

Substitute back

= \frac{1}{8} (- \frac{cos ^{7} ( 8 x )}{7})

Simplify

\frac{1}{8} (- \frac{cos ^{7} ( 8 x )}{7}) : - \frac{1}{56} cos^{7} (8 x)

= - \frac{1}{56} cos^{7} (8 x)

Add a constant to the solution

= - \frac{1}{56} cos^{7} (8 x) + C

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What is the integral of cos^6(8x)sin(8x) ?
The integral of cos^6(8x)sin(8x) is -1/56 cos^7(8x)+C