What is the integral of (cos(3x)sin(3x))/3 ?

The integral of (cos(3x)sin(3x))/3 is -1/18 cos^2(3x)+C

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integral of (cos(3x)sin(3x))/3

\int \frac{cos ( 3 x ) sin ( 3 x )}{3} d x

- \frac{1}{18} cos^{2} (3 x) + C

Solution steps

\int \frac{cos ( 3 x ) sin ( 3 x )}{3} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int cos (3 x) sin (3 x) d x

Apply u-substitution

: \int - \frac{u}{3} d u

= \frac{1}{3} \cdot \int - \frac{u}{3} d u

Take the constant out:

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

= \frac{1}{3} (- \frac{1}{3} \cdot \int u d u)

Apply the Power Rule

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

= \frac{1}{3} (- \frac{1}{3} \cdot \frac{u ^{2}}{2})

Substitute back

u = cos (3 x)

= \frac{1}{3} (- \frac{1}{3} \cdot \frac{cos ^{2} ( 3 x )}{2})

Simplify

\frac{1}{3} (- \frac{1}{3} \cdot \frac{cos ^{2} ( 3 x )}{2}) : - \frac{1}{18} cos^{2} (3 x)

= - \frac{1}{18} cos^{2} (3 x)

Add a constant to the solution

= - \frac{1}{18} cos^{2} (3 x) + C

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What is the integral of (cos(3x)sin(3x))/3 ?
The integral of (cos(3x)sin(3x))/3 is -1/18 cos^2(3x)+C