What is the integral of csc^6(x) ?

The integral of csc^6(x) is -(cos(x)csc^5(x))/5+4/5 (-1/3 csc^3(x)cos(x)-2/3 cot(x))+C

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integral of csc^6(x)

\int csc^{6} (x) d x

- \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} (- \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x)) + C

Solution steps

\int csc^{6} (x) d x

Apply Integral Reduction

= - \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} \cdot \int csc^{4} (x) d x

\int csc^{4} (x) d x = - \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x)

= - \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} (- \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x))

Add a constant to the solution

= - \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} (- \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x)) + C

What is the integral of csc^6(x) ?
The integral of csc^6(x) is -(cos(x)csc^5(x))/5+4/5 (-1/3 csc^3(x)cos(x)-2/3 cot(x))+C