Question

Evaluate the integral: ∫cosec³ x dx

Answer 1

It is given that

∫cosec³ x dx

We can write it as

∫cosec³ x dx = ∫cosec x cosec² x dx

So we get

By integration we get

= cosec x (- cot x) – ∫ (- cosec x cot x) (- cot x) dx

It can be written as

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

Here cot² x = cosec² x – 1

We get

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

By further simplification

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

We know that ∫ cosec³ x dx = 1

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

On further calculation

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

By integration we get

2 ∫cosec³ x dx = – cosec x cot x + log |tan x/2| + c

Here

∫cosec³ x dx = – 1/2 cosec x cot x + 1/2 log |tan x/2| + c