What is the integral of cos^{12}(7t)sin(7t) ?

The integral of cos^{12}(7t)sin(7t) is -1/91 cos^{13}(7t)+C

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integral of cos^{12}(7t)sin(7t)

\int cos^{12} (7 t) sin (7 t) d t

- \frac{1}{91} cos^{13} (7 t) + C

Solution steps

\int cos^{12} (7 t) sin (7 t) d t

Apply u-substitution

= \int cos^{12} (u) sin (u) \frac{1}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int cos^{12} (u) sin (u) d u

Apply u-substitution

= \frac{1}{7} \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}{7} (- \int v^{12} d v)

Apply the Power Rule

= \frac{1}{7} (- \frac{v ^{13}}{13})

Substitute back

= \frac{1}{7} (- \frac{cos ^{13} ( 7 t )}{13})

Simplify

\frac{1}{7} (- \frac{cos ^{13} ( 7 t )}{13}) : - \frac{1}{91} cos^{13} (7 t)

= - \frac{1}{91} cos^{13} (7 t)

Add a constant to the solution

= - \frac{1}{91} cos^{13} (7 t) + C

What is the integral of cos^{12}(7t)sin(7t) ?
The integral of cos^{12}(7t)sin(7t) is -1/91 cos^{13}(7t)+C