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

**Question:**

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

**Solution:**

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 = ∫ Sec****² **x

**v = tan x**

Formula:

**∫ 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

**Related pages**

**Integration****Example problems using he above formulas****Substitution method****Decomposition method****Properties of integrals****Integration-by parts****Standard integrals****Integrating quadratic denominator****Integration-using partial fractions****Definite integrals**

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