In this page Integral of secant cubed we are going to see how to integrating sec^3 x with detailed steps.


Integrate Sec^3 x with respect to x.  Integral of secant cubed


In the first step we are going to split Sec³ x as (sec x) (Sec² x).

∫ Sec³ x dx  = ∫ (Sec x) (Sec² x) dx

u = sec x                                      dv = Sec² x

du = sec x tan x dx                    ∫ dv = ∫ Sec² x   

                                                        v = tan x


∫ u dv  = uv - ∫ v du

∫ Sec³ x dx = (sec x) (tan x) - ∫ (tan x) (sec x tan x) dx

                = (sec x) (tan x) - ∫ (tan x)² (sec x) dx     

                = (sec x) (tan x) - ∫ (tan²x) (sec x) dx

tan²x can be written as sec ² x - 1

                = (sec x) (tan x) - ∫ (sec ² x - 1) (sec x) dx

Now we are multiplying sec x with sec ² x - 1

∫ Sec³ x dx = (sec x) (tan x) - ∫ (sec ³ x - sec x) dx

∫ Sec³ x dx = (sec x) (tan x) - ∫ sec ³ x dx + ∫ sec x dx

Now we are adding ∫ sec ³ x dx in order to cancel ∫ sec ³ x dx o the right side.

∫ Sec³ x dx + ∫ Sec³ x dx = (sec x) (tan x) + ∫ sec x dx

We have formula for integrating sec x that is log (sec x + tan x )

2 ∫ Sec³ x dx = (sec x) (tan x) + log (sec x + tan x ) + C

∫ Sec³ x dx = (1/2)(sec x) (tan x) + log (sec x + tan x ) + C

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Integral of Secant Cubed to Integration

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