Question

Evaluate: ∫ cosec³ x dx

Answer 1

Let I = ∫ cosec³ x dx 

= ∫ cosec³ x .cosec x dx 

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

= – cosec x .cot x – ∫ cosec x .cot² x dx 

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

= – cosec x .cot x – ∫ cosec³ x dx + ∫ cosec x dx 

2I = – cosec x .cot x + ∫ cosec x dx 

2I = – cosec x .cot x + log |cosec x – cot x| + c

I = (1/2) (– cosec x .cot x + log |cosec x – cot x| + c