What is the integral of cos(3x)sin^{10}(3x) ?

The integral of cos(3x)sin^{10}(3x) is 1/33 sin^{11}(3x)+C

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integral of cos(3x)sin^{10}(3x)

\int cos (3 x) sin^{10} (3 x) d x

\frac{1}{33} sin^{11} (3 x) + C

Solution steps

\int cos (3 x) sin^{10} (3 x) d x

Apply u-substitution

: \int cos (u) sin^{10} (u) \frac{1}{3} d u

= \int cos (u) sin^{10} (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int cos (u) sin^{10} (u) d u

Apply u-substitution

: \int v^{10} d v

= \frac{1}{3} \cdot \int v^{10} d v

Apply the Power Rule

: \frac{v ^{11}}{11}

= \frac{1}{3} \cdot \frac{v ^{11}}{11}

Substitute back

= \frac{1}{3} \cdot \frac{sin ^{11} ( 3 x )}{11}

Simplify

\frac{1}{3} \cdot \frac{sin ^{11} ( 3 x )}{11} : \frac{1}{33} sin^{11} (3 x)

= \frac{1}{33} sin^{11} (3 x)

Add a constant to the solution

= \frac{1}{33} sin^{11} (3 x) + C

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What is the integral of cos(3x)sin^{10}(3x) ?
The integral of cos(3x)sin^{10}(3x) is 1/33 sin^{11}(3x)+C