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

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Related Questions

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