Question

Evaluate: ∫ cot³ x . cosec³ x dx

Answer 1

Let cosec x = t

=  – cosec x .cot x dx = dt

= cosec x .cot x dx = – dt 

= ∫ cot³ x . cosec³ x dx 

= ∫ cot² x . cot x . cosec³ x dx 

= ∫ (cosec² x – 1). cosec² x (cosec x . cot x) dx 

= – ∫ (t² – 1)t² dt 

= ∫ (t⁴ – t²) dt 

= ((t⁵/ 5) – (t³/ 3)) + c 

= ((cosec⁵x/ 5) – (cosec³ x/ 3)) + c