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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Substitute back

u = cos (3 x)

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

Simplify

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

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

Add a constant to the solution

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

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