What is the integral of (csc(x))^4 ?

The integral of (csc(x))^4 is -1/3 csc^3(x)cos(x)-2/3 cot(x)+C

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integral of (csc(x))^4

\int (csc (x))^{4} d x

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

Solution steps

\int (csc (x))^{4} d x

Simplify

= \int csc^{4} (x) d x

Apply Integral Reduction

= - \frac{cos ( x ) csc ^{3} ( x )}{3} + \frac{2}{3} \cdot \int csc^{2} (x) d x

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

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (csc(x))^4 ?
The integral of (csc(x))^4 is -1/3 csc^3(x)cos(x)-2/3 cot(x)+C