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

The integral of cos(5x)sin(5x) is -1/10 cos^2(5x)+C

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

\int cos (5 x) sin (5 x) d x

- \frac{1}{10} cos^{2} (5 x) + C

Solution steps

\int cos (5 x) sin (5 x) d x

Apply u-substitution

: \int - \frac{u}{5} d u

= \int - \frac{u}{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 u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= - \frac{1}{5} \cdot \frac{u ^{2}}{2}

Substitute back

u = cos (5 x)

= - \frac{1}{5} \cdot \frac{cos ^{2} ( 5 x )}{2}

Simplify

- \frac{1}{5} \cdot \frac{cos ^{2} ( 5 x )}{2} : - \frac{1}{10} cos^{2} (5 x)

= - \frac{1}{10} cos^{2} (5 x)

Add a constant to the solution

= - \frac{1}{10} cos^{2} (5 x) + C

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What is the integral of cos(5x)sin(5x) ?
The integral of cos(5x)sin(5x) is -1/10 cos^2(5x)+C