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.

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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