What is the integral of-15cos^2(5x)sin(5x) ?

The integral of-15cos^2(5x)sin(5x) is cos^3(5x)+C

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integral of-15cos^2(5x)sin(5x)

\int - 15 cos^{2} (5 x) sin (5 x) d x

cos^{3} (5 x) + C

Solution steps

\int - 15 cos^{2} (5 x) sin (5 x) d x

Take the constant out:

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

= - 15 \cdot \int cos^{2} (5 x) sin (5 x) d x

Apply u-substitution

: \int cos^{2} (u) sin (u) \frac{1}{5} d u

= - 15 \cdot \int cos^{2} (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

= - 15 \cdot \frac{1}{5} \cdot \int cos^{2} (u) sin (u) d u

Apply u-substitution

: \int - v^{2} d v

= - 15 \cdot \frac{1}{5} \cdot \int - v^{2} d v

Take the constant out:

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

= - 15 \cdot \frac{1}{5} (- \int v^{2} d v)

Apply the Power Rule

: \frac{v ^{3}}{3}

= - 15 \cdot \frac{1}{5} (- \frac{v ^{3}}{3})

Substitute back

= - 15 \cdot \frac{1}{5} (- \frac{cos ^{3} ( 5 x )}{3})

Simplify

- 15 \cdot \frac{1}{5} (- \frac{cos ^{3} ( 5 x )}{3}) : cos^{3} (5 x)

= cos^{3} (5 x)

Add a constant to the solution

= cos^{3} (5 x) + C

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What is the integral of-15cos^2(5x)sin(5x) ?
The integral of-15cos^2(5x)sin(5x) is cos^3(5x)+C