Question

Evaluate ∫ {cotx - cot³}x/(1+cot³x) dx

\(\int\frac{\{cot\,x+cot^3\}x}{1+cot^3x}\)dx

Answer 1

∫ {cotx - cot³}x/(1+cot³x) dx

= \(\int\frac{cotx(1+cot^2x)}{1+cot^3x}\)dx

= \(\int\frac{cotx\,cosec^2x)}{1+cot^3x}\)dx

Put cot x = t, 

-cosec²x dx = dt;

= \(-\int\frac{tdt}{t^3+1}\)

= \(-\int\frac{tdt}{(t+1)(t^2-t+1)}\)

By partial fractions it’s a remembering thing,

That if you see the above integral just apply the below return result,