Here you will learn proof of integration of cosecx or cosec x and examples based on it.
Let’s begin –
Integration of Cosecx or Cosec x
The integration of cosec x is log |cosec x – cot x| + C or \(log |tan {x\over 2}|\) + C.
where C is the integration constant.
i.e. \(\int\) cosec x = log |cosec x – cot x| + C
or, \(\int\) cosec x = \(log |tan {x\over 2}|\) + C
Proof :
Let I = \(\int\) cosec x dx.
Multiply and divide both denominator and numerator by cosec x – cot x.
Then, I = \(\int\) \(cosec x(cosec x – cot x)\over (cosec x – cot x)\) dx
Let cosec x – cot x = t. Then,
d(cosec x – cot x) =dt
\(\implies\) \((-cosec x cot x + cosec^2 x)\) dx = dt
\(\implies\) dx = \({dt\over cosec x (cosec x – cot x)}\)
Putting cosec x – cot x = t and dx = \({dt\over sec x (sec x + tan x)}\), we get
I = \(\int\) \(sec x (sec x + tan x)\over t\) \(\times\) \({dt\over cosec x (cosec x – cot x)}\)
= \(\int\) \(1\over t\) dt = log | t | + C
= log |cosec x – cot x| + C
Hence, I = log |cosec x – cot 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 = \(2 sin^2 x\)
\(\implies\) I = \(1\over \sqrt{2sin^2 x}\)
\(\implies\) I = \(1\over \sqrt{2}\) \(1\over sin x\) dx
= \(1\over \sqrt{2}\) \(\int\) cosec x dx
= \(1\over \sqrt{2}\) log |cosec x – cot x| + C
Related Questions
What is the Differentiation of cosec x ?
What is the Integration of Sec Inverse x and Cosec Inverse x ?
