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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - u^{2} d u

= \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}

= - \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

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