What is the integral of \sqrt[7]{cot(x)}csc^2(x) ?

The integral of \sqrt[7]{cot(x)}csc^2(x) is -cot(x)\sqrt[7]{cot(x)}+1/8 cot^{8/7}(x)+C

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integral of \sqrt[7]{cot(x)}csc^2(x)

\int 7 cot (x) csc^{2} (x) d x

- cot (x) 7 cot (x) + \frac{1}{8} cot^{\frac{8}{7}} (x) + C

Solution steps

\int 7 cot (x) csc^{2} (x) d x

Apply Integration By Parts

: - cot (x) 7 cot (x) - \int \frac{1}{7} csc^{2} (x) cot^{\frac{1}{7}} (x) d x

= - cot (x) 7 cot (x) - \int \frac{1}{7} csc^{2} (x) cot^{\frac{1}{7}} (x) d x

\int \frac{1}{7} csc^{2} (x) cot^{\frac{1}{7}} (x) d x = - \frac{1}{8} cot^{\frac{8}{7}} (x)

= - cot (x) 7 cot (x) - (- \frac{1}{8} cot^{\frac{8}{7}} (x))

Simplify

= - cot (x) 7 cot (x) + \frac{1}{8} cot^{\frac{8}{7}} (x)

Add a constant to the solution

= - cot (x) 7 cot (x) + \frac{1}{8} cot^{\frac{8}{7}} (x) + C

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What is the integral of \sqrt[7]{cot(x)}csc^2(x) ?
The integral of \sqrt[7]{cot(x)}csc^2(x) is -cot(x)\sqrt[7]{cot(x)}+1/8 cot^{8/7}(x)+C