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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

= 2 (- \int u^{2} d u)

Apply the Power Rule

= 2 (- \frac{u ^{3}}{3})

Substitute back

u = cos (2 x)

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

Simplify

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

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

Add a constant to the solution

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

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