What is the integral of cos^9(5t)sin(5t) ?

The integral of cos^9(5t)sin(5t) is -1/50 cos^{10}(5t)+C

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integral of cos^9(5t)sin(5t)

\int cos^{9} (5 t) sin (5 t) d t

- \frac{1}{50} cos^{10} (5 t) + C

Solution steps

\int cos^{9} (5 t) sin (5 t) d t

Apply u-substitution

: \int cos^{9} (u) sin (u) \frac{1}{5} d u

= \int cos^{9} (u) sin (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int cos^{9} (u) sin (u) d u

Apply u-substitution

: \int - v^{9} d v

= \frac{1}{5} \cdot \int - v^{9} d v

Take the constant out:

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

= \frac{1}{5} (- \int v^{9} d v)

Apply the Power Rule

: \frac{v ^{10}}{10}

= \frac{1}{5} (- \frac{v ^{10}}{10})

Substitute back

= \frac{1}{5} (- \frac{cos ^{10} ( 5 t )}{10})

Simplify

\frac{1}{5} (- \frac{cos ^{10} ( 5 t )}{10}) : - \frac{1}{50} cos^{10} (5 t)

= - \frac{1}{50} cos^{10} (5 t)

Add a constant to the solution

= - \frac{1}{50} cos^{10} (5 t) + C

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What is the integral of cos^9(5t)sin(5t) ?
The integral of cos^9(5t)sin(5t) is -1/50 cos^{10}(5t)+C