What is the integral of cos^4(5x)sin(5x) ?

The integral of cos^4(5x)sin(5x) is -1/25 cos^5(5x)+C

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integral of cos^4(5x)sin(5x)

\int cos^{4} (5 x) sin (5 x) d x

- \frac{1}{25} cos^{5} (5 x) + C

Solution steps

\int cos^{4} (5 x) sin (5 x) d x

Apply u-substitution

: \int cos^{4} (u) sin (u) \frac{1}{5} d u

= \int cos^{4} (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^{4} (u) sin (u) d u

Apply u-substitution

: \int - v^{4} d v

= \frac{1}{5} \cdot \int - v^{4} 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^{4} d v)

Apply the Power Rule

: \frac{v ^{5}}{5}

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

Substitute back

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

Simplify

\frac{1}{5} (- \frac{cos ^{5} ( 5 x )}{5}) : - \frac{1}{25} cos^{5} (5 x)

= - \frac{1}{25} cos^{5} (5 x)

Add a constant to the solution

= - \frac{1}{25} cos^{5} (5 x) + C

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What is the integral of cos^4(5x)sin(5x) ?
The integral of cos^4(5x)sin(5x) is -1/25 cos^5(5x)+C