Question

Evaluate: ∫ sec³ x dx

Answer 1

Let I = ∫ sec³ x dx 

= ∫ (sec² x .sec x) dx 

= sec x ∫ sec² x dx – ∫ (sec x .tan x .tan x) dx 

= sec x ∫ sec² x dx – ∫ (sec x .tan² x) dx 

= sec x. tan x – ∫ (sec x (sec² x – 1)) dx 

= sec x. tan x – ∫ sec³ x dx + ∫ sec x dx 

2I = sec x .tan x + ∫ sec x dx 

2I = sec x .tan x – log |sec x + tan x| + c 

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