Question

Evaluate: ∫ sec⁵ x dx

Answer 1

Let I = ∫ sec⁵ x dx 

= ∫ sec³ x .sec² x dx 

= sec³ x ∫ sec² x dx – ∫ (3 sec³ x .tan x).tan x dx 

= sec³ x .tan x – 3 ∫ sec³ x (sec² x – 1) dx 

= sec³ x .tan x – 3 ∫ sec⁵ x dx + 3 ∫ sec³ x dx 

 = sec³ x .tan x – I – (1/2) (sec x .tan x + log |sec x + tan x|) + c

= 2I = sec³ x tan x – (1/2) (sec x tan x + log |sec x + tan x|) + c

= I = (1/2) (sec³ x tan x) – (1/4) (sec x tan x + log |sec x + tan x|) + c