Calculating integrals involving floor function

[smatik]

Is it possible to claculate ∫[cot(x)]dx from x=0 to x=$\pi$where [.] represents floor functon or the greatest integer function??
it seems impossible to me but can we use the properties of definite integrals to somehow evaluate the area?

 

[mfb]

It is possible to convert this to a series (involving arctan), but I have no idea if this series can be evaluated analytically. It is possible to get a numeric approximation, of course.

 

[JJacquelin]

Is it possible to claculate ∫[cot(x)]dx from x=0 to x=$\pi$where [.] represents floor functon or the greatest integer function?

Impossible because this defined integral is not converging.

 

[mfb]

Well, we can consider
$$\lim_{\delta \to 0} \int_\delta^{\pi-\delta} [\cot(x)] dx$$
This should be well-defined and finite.

 

[JJacquelin]

Yes, if both limits of the integral tend respectively to 0 and pi, with the same gap (delta), this is similar to the Cauchy integral (Principal Value). Then the limit of the value of the integral is finite : -pi/2 in case of the floor function and pi/2 in case of the ceiling function.

 

[mfb]

Right, we can use the symmetry of the function. Nice.