Question

Give reduction formula for ∫ secⁿ x dx.

Answer 1

Let I_n = ∫ secⁿ x dx 

= ∫ sec^{n– 2} x .sec² x dx 

= sec^{n– 2} x ∫ sec² x dx – ∫ (n – 2)sec^{n – 2} x .tan² x dx 

= sec^{n– 2} x tan x – (n – 2) ∫ sec^{n –2} x .(sec² x – 1)dx 

= sec^{n– 2} x tan x – (n – 2) ∫ (secⁿ x – sec^{n– 2} x)dx 

= sec^{n– 2} x tan x – (n – 2)I_n + (n – 2)I_n–2 

(n – 2)I_n = sec^n–2 x tan x – (n – 2)I_n–2 

I_n = ((sec^{n– 2} x tan x) /(n – 1)) + ((n – 2) /(n – 1))I_{n – 2} + C 

which is the required reduction formula.