# INTEGRATION OF SEC CUBE X

We can integrate sec3x using the method integration by integration by parts.

Integration by parts (formula) :

∫udv = uv - ∫vdu

∫sec3x dx = ∫sec x ⋅ sec2x dx

Let u = sec x and dv = sec2x.

 u = sec xᵈᵘ⁄dₓ = sec x ⋅ tan xdu = sec x ⋅ tan x ⋅ dx dv = sec2x∫dv = ∫sec2xv = tan x + C

Using the formula of integration by parts,

∫sec3x dx = sec x ⋅ tan x - tan x sec x tan x ⋅ dx

∫sec3x dx = sec x ⋅ tan x - tan2 x sec x ⋅ dx

∫sec3x dx = sec x ⋅ tan x - ∫(sec2 x - 1)sec x ⋅ dx

∫sec3x dx = sec x ⋅ tan x - ∫(sec3 x - sec x)dx

∫sec3x dx = sec x ⋅ tan x - ∫sec3 x dx + ∫sec x dx

Add ∫sec3 x dx to both sides.

2∫sec3x dx = sec x ⋅ tan x + ∫sec x dx

2∫sec3x dx = sec x ⋅ tan x + ln|sec x + tan x| + C

Divide both sides by 2.

∫sec3x dx = ½(sec x ⋅ tan x + ln|sec x + tan x|) + C

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