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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - u - 2 d u

= \int - u - 2 d u

Take the constant out:

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

= - \int u - 2 d u

Apply u-substitution

: \int v d v

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

Add a constant to the solution

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

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\int \frac{2 x ^{3} + 2 x ^{2} + 1}{x ^{2} - x} d x

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\int 7 x^{\frac{2}{5}} + 8 x^{\frac{- 4}{5}} d x

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