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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - 2 - u d u

= \int - 2 - u d u

Take the constant out:

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

= - \int 2 - u d u

Apply u-substitution

: \int - v d v

= - \int - v d v

Take the constant out:

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

= - (- \int v d v)

Apply the Power Rule

: \frac{2}{3} v^{\frac{3}{2}}

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

Substitute back

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

Simplify

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

Add a constant to the solution

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

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