What is the integral of csc^2(x)sqrt(1-3cot(x)) ?

The integral of csc^2(x)sqrt(1-3cot(x)) is 2/9 (1-3cot(x))^{3/2}+C

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integral of csc^2(x)sqrt(1-3cot(x))

\int csc^{2} (x) 1 - 3 cot (x) d x

\frac{2}{9} (1 - 3 cot (x))^{\frac{3}{2}} + C

Solution steps

\int csc^{2} (x) 1 - 3 cot (x) d x

Apply u-substitution

: \int - 1 - 3 u d u

= \int - 1 - 3 u d u

Take the constant out:

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

= - \int 1 - 3 u d u

Apply u-substitution

: \int - \frac{v}{3} d v

= - \int - \frac{v}{3} d v

Take the constant out:

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

= - (- \frac{1}{3} \cdot \int v d v)

Apply the Power Rule

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

= - (- \frac{1}{3} \cdot \frac{2}{3} v^{\frac{3}{2}})

Substitute back

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

Simplify

- (- \frac{1}{3} \cdot \frac{2}{3} (1 - 3 cot (x))^{\frac{3}{2}}) : \frac{2}{9} (1 - 3 cot (x))^{\frac{3}{2}}

= \frac{2}{9} (1 - 3 cot (x))^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{9} (1 - 3 cot (x))^{\frac{3}{2}} + C

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What is the integral of csc^2(x)sqrt(1-3cot(x)) ?
The integral of csc^2(x)sqrt(1-3cot(x)) is 2/9 (1-3cot(x))^{3/2}+C