hat is the integration of sec^3(x)

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Discussion Response Post to: hat is the integration of sec^3(x)

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Juhi Sharma (1527)

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let u = sec x dv = sec^2 x
then du = sec x tan x v = tan x

so int(sec^3 x dx) = sec x tan x - int(sec x tan^2 x dx)
= sec x tan x - int(sec x (sec^2 x - 1) dx)
= sec x tan x - int(sec^3 x dx) + int(sec x dx)
= sec x tan x - int(sec^3 x dx) + ln (sec x + tan x) + C

so now we have
int(sec^3 x dx) = sec x tan x - int(sec^3 x dx) + ln (sec x + tan x) + C

solve for int(sec^3 x dx):
2 int(sec^3 x dx) = sec x tan x + ln (sec x + tan x) + C
int(sec^3 x dx) = 1/2 (sec x tan x + ln (sec x + tan x)) + C

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