What is the integral of cot(θ)csc^n(θ) ?

The integral of cot(θ)csc^n(θ) is -1/n csc^n(θ)+C

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integral of cot(θ)csc^n(θ)

\int cot (θ) csc^{n} (θ) d θ

- \frac{1}{n} csc^{n} (θ) + C

Solution steps

\int cot (θ) csc^{n} (θ) d θ

Apply u-substitution

= \int - \frac{1}{n} d u

Integral of a constant:

\int a d x = a x

= (- \frac{1}{n}) u

Substitute back

u = csc^{n} (θ)

= (- \frac{1}{n}) csc^{n} (θ)

Simplify

= - \frac{1}{n} csc^{n} (θ)

Add a constant to the solution

= - \frac{1}{n} csc^{n} (θ) + C

What is the integral of cot(θ)csc^n(θ) ?
The integral of cot(θ)csc^n(θ) is -1/n csc^n(θ)+C