Can this be correct $\lim_{n\to \infty}\int_0^{\pi\over 2}\cos x\cos\left({x\over n }\right)\ln(\ln\csc x) \, dx = -\gamma?$

Question

$$\lim_{n\to \infty}\int_0^{\pi\over 2}\cos x\cos\left({x\over n } \right) \log(\log\csc (x)) \, dx$$

surprisingly give a result $-0.577156\ldots$, which went we check up it, it is Euler's constant value!!!.

Is this true or Wolfram alpha integration has a bug in its software?

I don't think using basic Integration by part or substitution will work to solve this problem. This is pass my knowledge. If this integral is correct, then it Just by luck we found on wolfram integral.

Answer 1

Yes, that's true!

First, observe that you can replace the $\cos(\frac{x}{n})$ term with $1$ by the Monotone convergence theorem.

Then, do the substitution $t=\sin(x)$, so $dt=\cos(x)dx$ and $t$ takes values from 0 to 1. We get

$\int_0^1 \log(\log(\frac{1}{t}))dt = \int_0^1 \log(-\log(t))dt$

The last integral is indeed $\gamma$, as one can see in Wikipedia.

Alternatively, one can make another substitution to get $\int_0^{\infty} e^{-t}\log(t)dt$ and look at this question to see a proof the value is $\gamma$.