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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int u^{2} d u

= \int u^{2} d u

Apply the Power Rule

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

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

Substitute back

u = 2 - cos (2 x)

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

Simplify

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

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

Add a constant to the solution

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

\frac{2}{s ^{2} - s}

8 cos (4 x), 8 - 8 cos (4 x), 0, \frac{π}{4}

\frac{d}{d x} (\frac{2 x}{2 + x})

\frac{d}{d x} (cos (\frac{1 - e ^{4 x}}{1 + e ^{4 x}}))

\int x cos (3 π x^{2}) d x

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