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 ?

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