What is the integral of sin(x)sqrt(1+2cos(x)) ?

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

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integral of sin(x)sqrt(1+2cos(x))

\int sin (x) 1 + 2 cos (x) d x

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

Solution steps

\int sin (x) 1 + 2 cos (x) d x

Apply u-substitution

: \int - \frac{u}{2} d u

= \int - \frac{u}{2} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int u d u

Apply the Power Rule

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

= - \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + 2 cos (x)

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

Simplify

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

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

Add a constant to the solution

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

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