What is the integral of sin^5(2x)cos^8(2x) ?

The integral of sin^5(2x)cos^8(2x) is -1/18 cos^9(2x)+1/11 cos^{11}(2x)-1/26 cos^{13}(2x)+C

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integral of sin^5(2x)cos^8(2x)

\int sin^{5} (2 x) cos^{8} (2 x) d x

- \frac{1}{18} cos^{9} (2 x) + \frac{1}{11} cos^{11} (2 x) - \frac{1}{26} cos^{13} (2 x) + C

Solution steps

\int sin^{5} (2 x) cos^{8} (2 x) d x

Apply u-substitution

: \int sin^{5} (u) cos^{8} (u) \frac{1}{2} d u

= \int sin^{5} (u) cos^{8} (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 sin^{5} (u) cos^{8} (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int (1 - cos^{2} (u))^{2} sin (u) cos^{8} (u) d u

Apply u-substitution

: \int - v^{8} (1 - v^{2})^{2} d v

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

Expand

- v^{8} (1 - v^{2})^{2} : - v^{8} + 2 v^{10} - v^{12}

= \frac{1}{2} \cdot \int - v^{8} + 2 v^{10} - v^{12} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (- \int v^{8} d v + \int 2 v^{10} d v - \int v^{12} d v)

\int v^{8} d v = \frac{v ^{9}}{9}

\int 2 v^{10} d v = \frac{2 v ^{11}}{11}

\int v^{12} d v = \frac{v ^{13}}{13}

= \frac{1}{2} (- \frac{v ^{9}}{9} + \frac{2 v ^{11}}{11} - \frac{v ^{13}}{13})

Substitute back

= \frac{1}{2} (- \frac{cos ^{9} ( 2 x )}{9} + \frac{2 cos ^{11} ( 2 x )}{11} - \frac{cos ^{13} ( 2 x )}{13})

Simplify

\frac{1}{2} (- \frac{cos ^{9} ( 2 x )}{9} + \frac{2 cos ^{11} ( 2 x )}{11} - \frac{cos ^{13} ( 2 x )}{13}) : - \frac{1}{18} cos^{9} (2 x) + \frac{1}{11} cos^{11} (2 x) - \frac{1}{26} cos^{13} (2 x)

= - \frac{1}{18} cos^{9} (2 x) + \frac{1}{11} cos^{11} (2 x) - \frac{1}{26} cos^{13} (2 x)

Add a constant to the solution

= - \frac{1}{18} cos^{9} (2 x) + \frac{1}{11} cos^{11} (2 x) - \frac{1}{26} cos^{13} (2 x) + C

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What is the integral of sin^5(2x)cos^8(2x) ?
The integral of sin^5(2x)cos^8(2x) is -1/18 cos^9(2x)+1/11 cos^{11}(2x)-1/26 cos^{13}(2x)+C