Calculate integral recurrence formula

Question

I'm trying to calculate recurrence formula for the following integral:

$$I= \int{\frac{1}{{\sin}^n(x)}dx}$$

While I have some ideas for even sinus degrees, have nothing to do for $n$ being odd.

Answer 1

$$I_n=\int{\frac{1}{{\sin}^n x}dx}=\int \csc^n x dx = -\int \csc^{n-2} d(\cot x) $$

Integrate by parts,

$$I_n=-\csc^{n-2}\cot x -(n-2)\int \csc^{n-2} \cot^2 x dx $$ $$=-\csc^{n-2}\cot x -(n-2)\int \csc^{n-2} (\csc^2 x -1)dx $$ $$=-\csc^{n-2}\cot x -(n-2)I_n + (n-2)I_{n-2} $$

Thus, the recurrence formula is

$$I_n = -\frac1{n-1}\csc^{n-2}\cot x + \frac{n-2}{n-1}I_{n-2}$$