# Integration of Cotx

Here you will learn proof of integration of cotx or cot x and examples based on it.

Let’s begin –

## Integration of Cotx or Cot x

The integration of cotx is  log |sin x| + C or – log |cosec x| + C

i.e. $$\int$$ (cotx) dx =  log |sin x| + C or,

$$\int$$ (cotx) dx = -log |cosec x| + C

Proof :

Let I = $$\int$$ (cot x) dx

Then, I = $$\int$$ $$cos x\over sin x$$ dx

Let sin x = t

Then, d(sin x) = dt $$\implies$$ cos x dx = dt

$$\implies$$ dx = $$dt\over cos x$$

Putting sin x = t, and dx = $$dt\over cos x$$, we get

I = $$\int$$ $$cos x\over sin x$$ $$\times$$ $$dt\over cos x$$

= $$\int$$ $$1\over t$$ dt =  log |t| + C

=  log |sin x| + C

And sin x = $$1\over cosec x$$

$$\implies$$ I = log |1/cosec x| + C = $$log |cosec^{-1} x|$$ + C = -log |cosec x| + C

Hence, $$\int$$ (cotx) dx = log |sin x| + C or, $$\int$$ (cotx) dx = -log |cosec x| + C

Example : Evaluate : $$\int$$ $$\sqrt{{1+cos 2x}\over {1-cos 2x}}$$ dx

Solution : We have,

I = $$\int$$ $$\sqrt{{1+cos 2x}\over {1-cos 2x}}$$ dx

By Trigonometry formulas,

1 – cos 2x = $$2sin^2 x$$ and 1 + cos 2x = $$2cos^2 x$$

$$\implies$$ I = $$\int$$ $$\sqrt{{2cos^2 x}\over {2sin^2 x}}$$ dx

$$\implies$$ I = $$\int$$ $${cos x}\over {sin x}$$ dx

{$$\because$$ $${cos x}\over {sin x}$$ = cot x }

$$\implies$$ I = $$\int$$ cot x dx

$$\implies$$ I = log |sin x| + C = – log |cosec x| + C

### Related Questions

What is the Differentiation of cot x ?

What is the Integration of cot inverse x ?

What is the Differentiation of cot inverse x ?