What is the integral of+cos(3t)sin(3t) ?

The integral of+cos(3t)sin(3t) is -1/6 cos^2(3t)+C

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integral of+cos(3t)sin(3t)

\int + cos (3 t) sin (3 t) d t

- \frac{1}{6} cos^{2} (3 t) + C

Solution steps

\int cos (3 t) sin (3 t) d t

Apply u-substitution

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

= \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} \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = cos (3 t)

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

Simplify

- \frac{1}{3} \cdot \frac{cos ^{2} ( 3 t )}{2} : - \frac{1}{6} cos^{2} (3 t)

= - \frac{1}{6} cos^{2} (3 t)

Add a constant to the solution

= - \frac{1}{6} cos^{2} (3 t) + C

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What is the integral of+cos(3t)sin(3t) ?
The integral of+cos(3t)sin(3t) is -1/6 cos^2(3t)+C