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 ?

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