Integration calculation help

[fm_com_28]

Dear ,
Could any one help me in olving the following integraton:
∫ln(1+a*cos(x)) dx , the integration limits is between pi and -pi

 

[RF-OM]

It looks like it should be 2*PI*ln(1+a*cos(x)). This is how my computer simplify your formula.

Best regards,
RF-OM

 

[fm_com_28]

Of course this is not true. as the integration then will be = 0 , which I am sure it is not true. By the way , what is the program do you use?

 

[RF-OM]

I agree with answer 0. This is what I thought for the first time. Then I insert your formula into MathCAD-13 and asked to simplify it. This program may provide right results and wrong as well. This is why I was not quite sure and unfortunately I have no time to investigate it in more details.

Best regards,
RF-OM

 

[RF-OM]

I checked a few math handbooks. There is no solution for this integral, but in one Russian famous and very old (1953) handbook I found something. Unfortunately here is imposible to use mathematical fonts and I need to use Word file. I attached it. It is not exactly what you ask, but similar and may be helpful.

Best regards,
RF-OM

 

[fm_com_28]

Thanks ver much,
But do you still don't know how to deduce this result ?

 

[RF-OM]

I did not deduce it, I found it in rare handbook. I am not a mathematician, math is just a good tool that I use in my work, but not more. 1 into parentheses can be a reason for zero result, but in math zero is not always real zero. Probably theory of limits can be used to solve such singularity, but I am too busy to investigate it deeper. It is not easy to find solution in the integrals tables from handbook and it is much longer to find or derive detailed solution. If I find something else I let you know.

Best regards,
RF-OM

 

[fm_com_28]

After you gave me this solution, I had an idea about using differentiation under integration which I think will help in finding this integral very easily.

 

[RF-OM]

Good Luck with this investigation!

 

[kalyanram]

Lets consider
\[J\left( a \right) = \int_0^\pi {\ln \left( {1 + a \cdot \cos \left( x \right)} \right)dx} \Rightarrow \dot J\left( a \right) = \int_0^\pi {\frac{{\cos \left( x \right)}}{{1 + a \cdot \cos \left( x \right)}}dx}\]

We know that -1?a?1.

Now find \[\dot J\left( a \right)\] and then integrate considering that : \[J\left( a \right) = J\left( a \right) - J\left( 0 \right) = \int_0^a {\dot J\left( a \right)da}\]

~Kalyan.