<s>[INST] Find the value of $\lim\limits_{n \to \infty}n\left(\left(\int_0^1\frac{1}{1+x^n}\,\mathrm{d}x\right)^n-\frac{1}{2}\right)$ 
Find the value of the limit $$\lim_{n \to \infty}n\left(\left(\int_0^1 \frac{1}{1+x^n}\,\mathrm{d}x\right)^n-\frac{1}{2}\right)$$

I can't solve the integral $\int_0^1 \mathrm{\frac{1}{1+x^n}}\,\mathrm{d}x$. But maybe the questions doesn't require solving the integral. 
Apparently the $\lim_{n \to \infty}(\int_0^1 \mathrm{\frac{1}{1+x^n}}\,\mathrm{d}x)^n$ should be $\frac{1}{2}$ for the question to make sense. That's all I know.
 [/INST] Hint. If one knows the digamma function $\psi=\Gamma'/\Gamma$, one may write, as $n \to \infty$,
$$
\begin{align}
\int_0^1\frac1{1+x^n}\:dx&=\int_0^1\frac{1-x^n}{1-x^{2n}}\:dx
\\\\&\stackrel{u=x^{2n}}=\frac1{2n}\int_0^1\frac{(1-u^{1/2})u^{1/(2n)-1}}{1-u}\:du
\\\\&=\frac1{2n}\left[\psi\left(\frac1{2n}+\frac12 \right)-\psi\left(\frac1{2n} \right) \right]
\\\\&=1-\frac{\ln 2}{n}+\frac{\pi ^2-6 \ln^2 2}{12n^2}+O\left(\frac{1}{n^3}\right)
\end{align}
$$ giving, as $n \to \infty$,
$$
n\ln \left(1-\frac{\ln 2}{n}+\frac{\pi ^2-6 \ln^2 2}{12n^2}+O\left(\frac{1}{n^3}\right) \right)=-\ln 2+\frac{\pi ^2-6 \ln^2 2}{24n}+O\left(\frac{1}{n^2}\right)
$$ 
that is

$$
n\left[\left(\int_0^1\frac1{1+x^n}\:dx \right)^n-\frac12\right]\to \frac{\pi ^2-6 \ln^2 2}{24}.
$$

</s>