Indefinite Integration of sec3 x

Indefinite Integration of sec^3 x

How to find the indefinite integration of sec3 x: example and its solution.

Example

Find the given indefinite integral. The integral of (sec^3 x) dx

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.

Pythagorean identities

(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.

Indefinite integration of sec x

Divide both sides by 2.

Then (given) = (1/2)(sec x tan x - ln |sec x + tan x|) + C.