please integral cot^7x

Question

please write procedure step by step..Thx

Answer 1

cot⁷(x)=cos⁷(x)/sin⁷(x)=cos⁶(x).cos(x)/sin⁷(x).

cos⁶(x)=(1-sin²(x))³=1-3sin²(x)+3sin⁴(x)-sin⁶(x).

Let s=sin(x) then ds=cos(x)dx.

∫cot⁷(x)dx=∫(1-3s²+3s⁴-s⁶)/s⁷)ds=∫(s^-7-3s^-5+3s^-3-1/s)ds,

∫cot⁷(x)dx=-s^-6/6+3s^-4/4-3s^-2/2+ln(s)+C where C is a constant.

∫cot⁷(x)dx=-⅙csc⁶(x)+¾csc⁴(x)-(3/2)csc²(x)+ln(sin(x))+C.