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

: \int - u^{2} d u

= 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

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

= 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

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\frac{d}{d r} (\frac{120 r}{r + 120})

x \to - 2 lim (x^{3} (5 - 3 x^{2}))

y^{^{'}} + 8 y + 18 = 0

\int 6 x - 4 d x

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