Find the integral: int sec^3 xdx

Anton Huynh 2022-11-20 Answered
Find the integral:
sec 3 x d x
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Answers (1)

Leo Robinson
Answered 2022-11-21 Author has 14 answers
Use Integration by Parts on sec 3 x d x
Let u = sec x , d v = sec 2 x , d u = sec x tan x d x , v = tan x
Substitute the above into u v v d u .
sec x tan x tan 2 x sec x d x
Use Pythagorean Identities: tan 2 x = sec 2 x 1
sec x tan x ( sec 2 x 1 ) sec x d x
Expand.
sec x tan x sec 3 x sec x d x
Use Sum Rule: f ( x ) + g ( x ) d x = f ( x ) d x + g ( x ) d x .
sec x tan x sec 3 d x + sec x d x
Set it as equal to the original integral sec 3 x d x
sec 3 x d x = sec x tan x sec 3 x d x + sec x d x
Add sec 3 x d x to both sides.
sec 3 x d x + sec 3 x d x = sec x tan x + sec x d x
Simplify sec 3 x d x + sec 3 x d x to 2 sec 3 x d x
2 sec 3 x d x = sec x tan x + sec x d x
Divide both sides by 2
sec 3 x d x = sec x tan x + sec x d x 2
Original integral solved.
sec x tan x + sec x d x 2
Use Trigonometric Integration: the integral of sec x
ln ( sec x + tan x )
sec x tan x + ln ( sec x + tan x ) 2
sec x tan x + ln ( sec x + tan x ) 2 + C
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