# Indefinite Integration of sec *x*

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

## Example

Multiply cos *x*

to both of the numerator and the denominator.

Set sin *x* = *t*.

Then cos *x* *dx* = *dt*.

Derivative of sin *x*

Substitute [sin *x*] with [*t*].

And substitute [cos *x* *dx*] with [*dt*].

Then (given) = ∫ 1/(1 - *t*^{2}) *dt*.

Integration by substitution (Part 1)

Factoring the difference of squares

Change the function to make (constant)/(linear) form

by using partial fraction decomposition.

Set 1/(1 + *t*)(1 - *t*) = *A*/(1 - *t*) + *B*/(1 - *t*)

and compare both sides' numerators

to find *A* and *B*.*A* = *B* = 1/2

So 1/(1 + *t*)(1 - *t*) = (1/2)(1/[1 - *t*]) + (1/2)(1/[1 - *t*])

= (1/2)(1/[1 - *t*] + 1/[1 - *t*]).

So (given) = ∫ (1/2)(1/[1 - *t*] + 1/[1 - *t*]) *dt*.

Integrate the integral.

Don't forget to change the sign of the latter term,

due to the (-) in (1 - *t*).

Linear change of variable rule

Indefinite integration of 1/*x*

Put sin *x* in *t*-s.

Then (given) = (1/2) ln |(1 + sin *x*)/(1 - sin *x*)| + *C*.

This is the answer.

But let's go further to see the practical formula.

Multiply (1 + sin *x*)/(1 + sin *x*).

Denominator

(1 - sin *x*)(1 + sin *x*) = 1 - sin^{2} *x*

Product of a sum and a difference

Take the square exponents

out from both of the numerator and the denominator.

Power of a quotient

Take the exponent out from the ln.

Logarithms of powers

Split the fraction into two parts.

1/cos *x* = sec *x*

sin *x*/cos *x* = tan *x*

So ∫ sec *x* *dx* = ln |sec *x* + tan *x*| + *C*.

This answer is the practical formula

that is widely used in various fields.