# Integration of Secx

Here you will learn proof of integration of secx or sec x and examples based on it.

Let’s begin –

## Integration of Secx or Sec x

The integration of sec x is log |sec x + tan x| + C or $$log |tan ({\pi\over 4} + {x\over 2})|$$ + C.

where C is the integration constant.

i.e. $$\int$$ sec x = log |sec x + tan x| + C

or, $$\int$$ sec x = $$log |tan ({\pi\over 4} + {x\over 2})|$$ + C

Proof :

Let I = $$\int$$ sec x dx.

Multiply and divide both denominator and numerator by sec x + tan x.

Then, I = $$\int$$ $$sec x(sec x + tan x)\over (sec x + tan x)$$ dx

Let sec x + tan x = t. Then,

d(sec x + tan x) =dt

$$\implies$$ $$(sec x tan x + sec^2 x)$$ dx = dt

$$\implies$$ dx = $${dt\over sec x (sec x + tan x)}$$

Putting sec x + tan x = t and dx = $${dt\over sec x (sec x + tan x)}$$, we get

I = $$\int$$ $$sec x (sec x + tan x)\over t$$ $$\times$$ $${dt\over sec x (sec x + tan x)}$$

= $$\int$$ $$1\over t$$ dt = log | t | + C

= log |sec x + tan x| + C

Hence, I = log |sec x + tan x| + C

Example : Evaluate $$1\over \sqrt{1 + cos 2x}$$ dx.

Solution : We have,

I = $$1\over \sqrt{1 + cos 2x}$$

By using differentiation formula, 1 + cos 2x = $$2 cos^2 x$$

$$\implies$$ I = $$1\over \sqrt{2cos^2 x}$$

$$\implies$$ I = $$1\over \sqrt{2}$$ $$1\over cos x$$ dx

= $$1\over \sqrt{2}$$ $$\int$$ sec x dx

= $$1\over \sqrt{2}$$ log |sec x + tan x| + C

### Related Questions

What is the Differentiation of sec x ?

What is the Integration of Sec Inverse x and Cosec Inverse x ?

What is the Differentiation of sec inverse x ?