# Find the ∫sec^3 xdx?

Find the integral of ∫sec^3 xdx

I need the steps used and answer please. I'll give 10 points to the best answer!

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- J DLv 51 decade agoFavorite Answer
∫sec^3 x dx

∫sec x sec^2 x dx

let u = sec x; dv = sec^2 x dx

du = sec x tan x; v = tan x

sec x tan x - ∫tan x sec x tan x dx

sec x tan x - ∫sec x tan^2 x dx

sec x tan x - ∫sec x (sec^2 x - 1) dx

so at this point we have...

∫sec^3 x dx = sec x tan x - ∫sec^3 x dx - ∫sec x dx

2∫sec^3 x dx = sec x tan x - ∫sec x dx

2∫sec^3 x dx = sec x tan x - ln|sec x + tan x| + c

∫sec^3 x dx = 1/2 sec x tan x - 1/2 ln|sec x + tan x| + c

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