What is the integral of sin(2x)(cos(2x))^{15} ?

The integral of sin(2x)(cos(2x))^{15} is -1/32 cos^{16}(2x)+C

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integral of sin(2x)(cos(2x))^{15}

\int sin (2 x) (cos (2 x))^{15} d x

- \frac{1}{32} cos^{16} (2 x) + C

Solution steps

\int sin (2 x) (cos (2 x))^{15} d x

Apply u-substitution

: \int \frac{1}{2} cos^{15} (u) sin (u) d u

= \int \frac{1}{2} cos^{15} (u) sin (u) 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^{15} (u) sin (u) d u

Apply u-substitution

: \int - v^{15} d v

= \frac{1}{2} \cdot \int - v^{15} 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^{15} d v)

Apply the Power Rule

: \frac{v ^{16}}{16}

= \frac{1}{2} (- \frac{v ^{16}}{16})

Substitute back

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

Simplify

\frac{1}{2} (- \frac{cos ^{16} ( 2 x )}{16}) : - \frac{1}{32} cos^{16} (2 x)

= - \frac{1}{32} cos^{16} (2 x)

Add a constant to the solution

= - \frac{1}{32} cos^{16} (2 x) + C

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What is the integral of sin(2x)(cos(2x))^{15} ?
The integral of sin(2x)(cos(2x))^{15} is -1/32 cos^{16}(2x)+C