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 August 1st, 2012, 10:21 PM #1 Newbie   Joined: Aug 2012 Posts: 11 Thanks: 0 Integration of log(cos(x))? Can anyone please tell how to calculate integration of log(cosx)?
 August 1st, 2012, 10:38 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs Re: Integration of log(cos(x))? I have split and moved your post here for now. I will wait for someone more knowledgeable to say whether this should be moved perhaps to the complex analysis forum.
 August 1st, 2012, 11:46 PM #3 Senior Member   Joined: Aug 2011 Posts: 333 Thanks: 8 Re: Integration of log(cos(x))? Hi ! The primitives of ln(cos(x)) cannot be expressed as a finite combination of usual functions. The analytic expression is complicated and includes a special function (polylogarithm).
 August 2nd, 2012, 02:34 AM #4 Newbie   Joined: Aug 2012 Posts: 11 Thanks: 0 Re: Integration of log(cos(x))? I'm not having much idea of polylogarithm function. It will be really helpful if we can get its solution. Thanks...
 August 2nd, 2012, 02:44 AM #5 Senior Member   Joined: Aug 2011 Posts: 333 Thanks: 8 Re: Integration of log(cos(x))?
 August 2nd, 2012, 03:06 AM #6 Senior Member   Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0 Re: Integration of log(cos(x))? It is possible to exactly calculate the definite integral of this if you go from one multiple of pi/2 (including zero) to another multiple of pi/2 and requires no special functions or complex numbers.
August 2nd, 2012, 03:45 AM   #7
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Re: Integration of log(cos(x))?

$\frac{i x^2}{2} - x \log$$1 + e^{i2x}$$ + x \log \, \cos \, x + \frac{i}{2} li_2$$-e^{i2x}$$$

Quote:
 Originally Posted by Najam I'm not having much idea of polylogarithm function.it will be really helpfull if we can get its solution.thanks...
Don't think that you can understand the solution without learning polylogarithm.

So, read it from wiki . . .

 August 2nd, 2012, 03:50 AM #8 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Integration of log(cos(x))? I think it is possible to calculate $\int_{0}^{\pi/2} \ln(\cos(x)) dx$ using complex analysis and using no special functions.
August 2nd, 2012, 04:11 AM   #9
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Re: Integration of log(cos(x))?

Quote:
 Originally Posted by Najam I'm not having much idea of polylogarithm function.it will be really helpfull if we can get its solution.thanks...

What everyone is trying to say is that this does not have an anti-derivative. That is, no nice closed form expression for the indefinite integral.

Add some limits of integration, then it is a rather famous integral. You can do a search and find it plenty of times.

As Fool said, the standard is usually $[0,\frac{\pi}{2}]$.

For more challenging problem, limits of integration such as $[0,\frac{\pi}{4}]$ may sometimes be used instead.

Just for fun, I ran it through Maple and it gave me

$1/2\,i{x}^{2}+x\ln \left( \cos \left( x \right) \right) -x\ln\left( 1+{e^{2\,ix}} \right) +1/2\,i{\it \text{polylog}} \left( 2,-{e^{2\,ix}} \right)$. Wolfram gives something similar.

The polylog in this case is specifically the 'dilogarithm' because of the power of n is 2 in the denominator of the sum below. Google it and you can see what it is.

$\text{polylog}(2,-e^{2ix})=\sum_{n=1}^{\infty}\frac{(-1)^{n}(e^{2ix})^{n}}{n^{2}}$

It is a special advanced function.

But, to put it into a more hands-on approach. if you did want to try your hand at it, perhaps begin by using $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$.

Break it down into several terms using the log laws, then apply the series for ln(1+x)

 August 2nd, 2012, 06:26 PM #10 Senior Member   Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0 Re: Integration of log(cos(x))? Here is how Maxima likes to display $\int \log(\cos x)dx$ : http://img600.imageshack.us/img600/7475/maxima1b.jpg If I didn't expand it it would simply put everything under a denominator of 2.

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