Integration of Cosecx

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 ?

What is the Differentiation of cosec inverse x ?

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