harmonic numbers $H_n=\sum_{i=1}^{n}\frac{1}{i}$
Euler's constant
$\gamma=\lim_{n\to \infty} [H_n-\ln(n)]$
Factorial
$n!=n(n-1)(n-2)\cdots2\cdot1$; valid for all non-negative integers
Show that,
$$(s+1)\gamma-\int_0^1\left(\frac{1}{\ln(x)}+\frac{1}{1-x}\right)\sum_{n=0}^{s}x^ndx=\sum_{i=1}^{s}H_i-\ln(s+1)!$$
Inspiring from the integral of Euler's constant; see Wikipedia $$\int_0^1\left(\frac{1}{\ln(x)}+\frac{1}{1-x}\right)=\gamma$$ Mathematical experimental
We did trial and error using wolfram integrator and observe the numerical values of the integral and we was able to come up with a closed form for it.
Unfortunately we are unable to provide a proof for it, can anyone provide us a prove of it?