What is the integral of cos^{12}(3t)sin(3t) ?

The integral of cos^{12}(3t)sin(3t) is -1/39 cos^{13}(3t)+C

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integral of cos^{12}(3t)sin(3t)

\int cos^{12} (3 t) sin (3 t) d t

- \frac{1}{39} cos^{13} (3 t) + C

Solution steps

\int cos^{12} (3 t) sin (3 t) d t

Apply u-substitution

: \int cos^{12} (u) sin (u) \frac{1}{3} d u

= \int cos^{12} (u) sin (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^{12} (u) sin (u) d u

Apply u-substitution

: \int - v^{12} d v

= \frac{1}{3} \cdot \int - v^{12} d v

Take the constant out:

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

= \frac{1}{3} (- \int v^{12} d v)

Apply the Power Rule

: \frac{v ^{13}}{13}

= \frac{1}{3} (- \frac{v ^{13}}{13})

Substitute back

= \frac{1}{3} (- \frac{cos ^{13} ( 3 t )}{13})

Simplify

\frac{1}{3} (- \frac{cos ^{13} ( 3 t )}{13}) : - \frac{1}{39} cos^{13} (3 t)

= - \frac{1}{39} cos^{13} (3 t)

Add a constant to the solution

= - \frac{1}{39} cos^{13} (3 t) + C

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What is the integral of cos^{12}(3t)sin(3t) ?
The integral of cos^{12}(3t)sin(3t) is -1/39 cos^{13}(3t)+C