What is the integral of sqrt(cos(2x))sin(2x) ?

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

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integral of sqrt(cos(2x))sin(2x)

\int cos (2 x) sin (2 x) d x

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

Solution steps

\int cos (2 x) sin (2 x) d x

Apply u-substitution

= \int - u^{2} d u

Take the constant out:

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

= - \int u^{2} d u

Apply the Power Rule

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

Substitute back

u = cos (2 x)

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

Simplify

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

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

Add a constant to the solution

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

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