Indefinite Integration of sec3 x
How to find the indefinite integration of sec3 x: example and its solution.
sec3 x = (sec x)(sec2 x)
Then use the integration by parts.
Set u = sec x and v' = sec2 x.
Write u = sec x.
Write u' = sec x tan x.
Derivative of sec x
Write v' = sec2 x next to u' = sec x tan x.
And write v = tan x next to u = sec x.
Indefinite integration of sec2 x
The integral of, uv', (sec x)(sec2 x) dx
is equal to,
uv, (sec x)(tan x)
minus the integral of, u'v, (sec x tan x)(tan x) dx.
(given) = sec x tan x - ∫ (sec x) dx - ∫ (sec3 x) dx
∫ (sec3 x) dx is the (given).
Then, instead of using the integration by parts again,
write the equation like this:
(given) = sec x tan x - ∫ (sec x) dx - (given).
Move the right side's (given) to the left side.
And write +C on the right side.
Divide both sides by 2.
Then (given) = (1/2)(sec x tan x - ln |sec x + tan x|) + C.