# Integeration of sec^3x w.r.t x

Godfrey Classic Prince
633 Points
13 years ago

Dear shakti kumar,

Here is the step by step procedure of integrating sec3x with respect to x

let u = sec x dv = sec^2 x
then du = sec x tan x v = tan x

so int(sec^3 x dx) = sec x tan x - int(sec x tan^2 x dx)
= sec x tan x - int(sec x (sec^2 x - 1) dx)
= sec x tan x - int(sec^3 x dx) + int(sec x dx)
= sec x tan x - int(sec^3 x dx) + ln (sec x + tan x) + C

so now we have
int(sec^3 x dx) = sec x tan x - int(sec^3 x dx) + ln (sec x + tan x) + C

solve for int(sec^3 x dx):
2 int(sec^3 x dx) = sec x tan x + ln (sec x + tan x) + C
int(sec^3 x dx) = 1/2 (sec x tan x + ln (sec x + tan x)) + C

note that the end result C is different from the above C, but since it's an arbitrary constant it doesn't matter !
This is the end result !!
Hope this helped you immensely..!!
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rampalli shyam
4 Points
13 years ago

1/3sec^3xtan^3x