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 August 1st, 2012, 10:21 PM #1 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 Global Moderator     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,566 Thanks: 108 Math Focus: The calculus 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: 306 Thanks: 1 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 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: 306 Thanks: 1 Re: Integration of log(cos(x))?
 August 2nd, 2012, 03:06 AM #6 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: 3,815 Thanks: 42 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 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.
August 2nd, 2012, 06:57 PM   #11

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Re: Integration of log(cos(x))?

Quote:
 Originally Posted by mathbalarka I think it is possible to calculate $\int_{0}^{\pi/2} \ln(\cos(x)) dx$ using complex analysis and using no special functions.
Yep, this is one I solved in high school, although I was given the hint that I could add something to it and use trig substitutions. However, it's easier than complex numbers. I was trying to find the continuous geometric mean under the curve sin x from 0 to pi and this integral came up. Here is how it's done:

$I=\int_{0}^{\pi/2} \log(\cos x) dx$
$2I=\int_{0}^{\pi/2} \log(\cos x) dx+\int_{0}^{\pi/2} \log(\sin x) dx$
$2I=\int_{0}^{\pi/2} \log(\cos x \sin x) dx$
At this point you substitute $\sin x \cos x=\frac{1}{2}\sin 2x$.
$2I=\int_{0}^{\pi/2} \log$$\frac{1}{2}\sin 2x$$ dx$
Using the property of logarithms you can remove the 1/2 and integrate the constant.
$2I=\int_{0}^{\pi/2} \log$$\sin 2x$$ dx-\frac{\pi}{2}\log 2$
Substitute x=u/2.
$2I=\frac{1}{2}\int_{0}^{\pi} \log$$\sin u$$ du-\frac{\pi}{2}\log 2$
If you take a close look at the integral you can see how it equals the original integral, I. This will allow you to solve for I.
$2I=I-\frac{\pi}{2}\log 2$
$I=-\frac{\pi}{2}\log 2$

This allowed me to show that the continuous geometric mean under sin x from 0 to pi is 1/2.

 August 2nd, 2012, 10:51 PM #12 Joined: Aug 2012 Posts: 11 Thanks: 0 Re: Integration of log(cos(x))? Thanks for explaining the solution of this integral, but these links which my friends have posted are giving directly the answer. I am interested in steps involved for its solution, because I need its approach to solve it, which will be helpful for the solution of other such type of integrals. Thanks..... :P
 August 2nd, 2012, 10:55 PM #13 Joined: Aug 2012 Posts: 11 Thanks: 0 Re: Integration of log(cos(x))? We want to calculate it by using indefinite integration because using definite integration and solving it is a cakewalk, by using property. So hope you'll help me in its solution. Thanks.
August 3rd, 2012, 03:08 AM   #14
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Re: Integration of log(cos(x))?

Quote:
 Originally Posted by Najam . . . links which my friends have posted are giving directly the answer.
Could you give the link to us.

Quote:
 Originally Posted by Najam We want to calculate it by using indefinite integration.
$\log$$\cos(x)$$$ doesn't have an anti-derivative in the logarithmic integral C(x, ln(x)).

Have you heard of Nonelementary Integrals?

This is a non-elementary integral and it is provable by Differential Galois Theory.

Balarka

.

 August 3rd, 2012, 03:50 AM #15 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,675 Thanks: 7 Re: Integration of log(cos(x))? [color=#000000]With complex analysis we may follow the same steps presented here.[/color]

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