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 x ᵈᵘ⁄dₓ = sec x ⋅ tan x du = sec x ⋅ tan x ⋅ dx |
dv = sec2x ∫dv = ∫sec2x v = 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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