If Zn= integration of (cot x)^n (n>1), show that Zn=-(cot x)^(n-1) / (n-1) - Z(n-2).

n > 1

so (cot x)^n = (cot x)^(n-2) cot ^2 x n- 2>= 0
= (cot x)^(n-2) (cosec ^2 x - 1)
= (cot x)^(n-2) cosec ^2 x - (cot x)^(n-2)

so
int (cot x)^n = int (cot x)^(n-2) cosec ^2 x - int (cot x)^(n-2)

if cot x = t then - cosec ^2 x dx = dt

so int (cot x)^(n-2) cosec ^2 x = - t^(n-2) dt = - t^(n-1)/(n-1) as n-2 >= 0 = - (cot x)^(n-1)/(n-1)

so int (cot x)^n = - (cot x)^(n-1)/(n-1) - int (cot x)^(n-2)

hence Zn=-(cot x)^(n-1) / (n-1) - Z(n-2)

QED

Z(n) +Z(n-2)
=Integral {(cotx)^n+(cotx)^(n-2)dx}
=Integral {(cotx)^(n-2) (cot^2(x)+1)dx}
=Integral {(cotx)^(n-2) cosec^2(x) dx}
=Integral {(cotx)^(n-2) (-1) d(cot(x)) dx}
=-(cotx)^(n-1) / (n-1) + C