# 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 ?