What is the integral of cos^7(2x)sin(2x) ?

The integral of cos^7(2x)sin(2x) is -1/16 cos^8(2x)+C

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integral of cos^7(2x)sin(2x)

\int cos^{7} (2 x) sin (2 x) d x

- \frac{1}{16} cos^{8} (2 x) + C

Solution steps

\int cos^{7} (2 x) sin (2 x) d x

Apply u-substitution

: \int cos^{7} (u) sin (u) \frac{1}{2} d u

= \int cos^{7} (u) sin (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int cos^{7} (u) sin (u) d u

Apply u-substitution

: \int - v^{7} d v

= \frac{1}{2} \cdot \int - v^{7} d v

Take the constant out:

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

= \frac{1}{2} (- \int v^{7} d v)

Apply the Power Rule

: \frac{v ^{8}}{8}

= \frac{1}{2} (- \frac{v ^{8}}{8})

Substitute back

= \frac{1}{2} (- \frac{cos ^{8} ( 2 x )}{8})

Simplify

\frac{1}{2} (- \frac{cos ^{8} ( 2 x )}{8}) : - \frac{1}{16} cos^{8} (2 x)

= - \frac{1}{16} cos^{8} (2 x)

Add a constant to the solution

= - \frac{1}{16} cos^{8} (2 x) + C

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What is the integral of cos^7(2x)sin(2x) ?
The integral of cos^7(2x)sin(2x) is -1/16 cos^8(2x)+C