In this question we encounter the Cauchy principal value of integrals which led me to ask for the value of this integral
$$i_n = PV \int_{0}^{\pi} x^n \tan(x)\,dx, n=0,1,2,...\tag{1}$$
Where the principal value in defined as
$$PV \int_{0}^{\pi}f(x)\,dx = \lim_{\epsilon\to +0}\left(\int_{0}^{\frac{\pi}{2}-\epsilon}f(x)\,dx+\int_{\frac{\pi}{2}+\epsilon}^{\pi}f(x)\,dx \right)\tag{2}$$
The first few values in the format $\{n,i_n\} $ are
$$\begin{align}\{0, 0\}, \{1, \pi \log(2)\}, \{2, -\pi^2 \log(2)\}, \{3, \pi^3 \log(2) -\frac{9}{8} \pi \zeta(3)\},\\ \{4, -\pi^4 \log(2) + \frac{9}{4} \pi^2 \zeta(3)\},\\ \{5, \pi^5 \log(2) - \frac{15}{4} \pi^3 \zeta(3) + \frac{225}{32} \pi \zeta(5)\}\end{align}\tag{3}$$
Here $\zeta(.)$ is the Riemann zeta function.
Questions
prove $(3)$
derive a general formula for $i_n$
Hint
Consider the generating integral
$$g(t) = PV \int_{0}^{\pi} e^{-t x} \tan(x)\,dx\tag{4}$$