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

The integral of sqrt(cos(3t))sin(3t) is -2/9 cos^{3/2}(3t)+C

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

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

- \frac{2}{9} cos^{\frac{3}{2}} (3 t) + C

Solution steps

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

Apply u-substitution

: \int - \frac{2 u ^{2}}{3} d u

= \int - \frac{2 u ^{2}}{3} d u

Take the constant out:

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

= - \frac{2}{3} \cdot \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

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

Substitute back

u = cos (3 t)

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

Simplify

- \frac{2}{3} \cdot \frac{( cos ( 3 t ) ) ^{3}}{3} : - \frac{2}{9} cos^{\frac{3}{2}} (3 t)

= - \frac{2}{9} cos^{\frac{3}{2}} (3 t)

Add a constant to the solution

= - \frac{2}{9} cos^{\frac{3}{2}} (3 t) + C

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What is the integral of sqrt(cos(3t))sin(3t) ?
The integral of sqrt(cos(3t))sin(3t) is -2/9 cos^{3/2}(3t)+C