# Indefinite Integration of sec^{3} *x*

How to find the indefinite integration of sec^{3} *x*: example and its solution.

## Example

sec^{3} *x* = (sec *x*)(sec^{2} *x*)

Then use the integration by parts.

Set *u* = sec *x* and *v*' = sec^{2} *x*.

Write *u* = sec *x*.

Write *u*' = sec *x* tan *x*.

Derivative of sec *x*

Write *v*' = sec^{2} *x* next to *u*' = sec *x* tan *x*.

And write *v* = tan *x* next to *u* = sec *x*.

Indefinite integration of sec^{2} *x*

The integral of, *u**v*', (sec *x*)(sec^{2} *x*) *dx*

is equal to,*u**v*, (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* - ∫ (sec^{3} *x*) *dx*

∫ (sec^{3} *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*.