Explain how to compute this integral?

[lucidicblur]

Can someone please explain how to compute this integral? This is not for school; I just came across it and I'm not sure what to do. Parts and various identities didn't help. $$\int$$ x/tan(x) dx

 

[slider142]

Can someone please explain how to compute this integral? This is not for school; I just came across it and I'm not sure what to do. Parts and various identities didn't help. $$\int$$ x/tan(x) dx

Parts should help. Let u(x) = x and let dv(x) = 1/tan(x). Then v(x) = ln|sin(x)| + C and du(x) = 1.

 

[lucidicblur]

Parts should help. Let u(x) = x and let dv(x) = 1/tan(x). Then v(x) = ln|sin(x)| + C and du(x) = 1.

This only results in having to integrate ln|sin(x)|, which does not help. I'm pretty sure parts is not the way to go.

 

[Hootenanny]

This only results in having to integrate ln|sin(x)|, which does not help. I'm pretty sure parts is not the way to go.

Now use integration by parts again with u(x) = ln|sin(x)| and dv(x) = 1. (You'll find that you'll have to use integration by parts once again after this step).

 

[lucidicblur]

Now use integration by parts again with u(x) = ln|sin(x)| and dv(x) = 1. (You'll find that you'll have to use integration by parts once again after this step).

after the first integration by parts i get xln|sin(x)|-$$\int$$ln|sin(x)|dx

using parts again, with u(x)=ln|sin(x)|, i get xln|sin(x)|-xln|sin(x)|-$$\int$$x/tanx dx

this is exactly where i started.

I entered this in an online integrator and got this as the answer:

xln(1-e^2ix)-1/2i(x²+Li₂(e^2ix)

which does not make sense to me.

 

[Hootenanny]

after the first integration by parts i get xln|sin(x)|-$$\int$$ln|sin(x)|dx

using parts again, with u(x)=ln|sin(x)|, i get xln|sin(x)|-xln|sin(x)|-$$\int$$x/tanx dx

this is exactly where i started.

Hmm, indeed this isn't really helpful. Initially, I thought that we could evaluate this integral using recurrence methods. However, the result you obtained from the online integrator suggests that the anti-derivative cannot be written in terms of elementary functions.

I entered this in an online integrator and got this as the answer:

xln(1-e^2ix)-1/2i(x²+Li₂(e^2ix)

which does not make sense to me.

Li(x) the so-called logarithmic integral function and is a special function. See http://en.wikipedia.org/wiki/Logarithmic_integral_function for more information.

 

[gabbagabbahey]

Hmm, indeed this isn't really helpful. Initially, I thought that we could evaluate this integral using recurrence methods. However, the result you obtained from the online integrator suggests that the anti-derivative cannot be written in terms of elementary functions.

Li(x) the so-called logarithmic integral function and is a special function. See http://en.wikipedia.org/wiki/Logarithmic_integral_function for more information.

Actually, in this case $$L_i(z)$$ represents the PolyLogarithm Function...which seems to suggest that perhaps it can be integrated by writing tanx in terms of complex exponentials and using a complex analysis method such as method of residues. Although I haven't tried it yet myself, it seems like a reasonable approach given the form of the solution.

 

[lucidicblur]

Actually, in this case $$L_i(z)$$ represents the PolyLogarithm Function...which seems to suggest that perhaps it can be integrated by writing tanx in terms of complex exponentials and using a complex analysis method such as method of residues. Although I haven't tried it yet myself, it seems like a reasonable approach given the form of the solution.

Thank you. This seems more probable. Is there anyway you can show me how to actually integrate this function or suggest any sites that would show how to evaluate these type of integrals.