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 ?