Related to the famous problem of the fastest path between two points (Brachistochrone) I created the following integral $$I(k)=\int_{0}^{1}\sqrt{\frac{x^{2(k-1)}+(1-x^k)^{2(1-k)/k}}{x^{2(k-1)}[1-(1-x^{k})^{1/k}]}}dx=\int_{0}^{1} f(x) dx.~~~~(1)$$ The simple interesting cases are $I(0)=1$, $I(1)=2\sqrt{2}$. When $k>1$ $$I(k)=\int_{0} \frac{dx}{x^{(3k-2)/2}}<\infty, ~ if~ 1<k<4/3.~~~~~~~(2)$$ NIntegrate of Mathematica gives $$I(0\le k,<k_c)=1~~~~(3)$$ and $$I(k_c<k<k_s)=3~~~~~~(4)$$ without warning any error and shows $I(k>k_s)$ as divergent, where $k_s=1.22$. $I(k)$ for $k \in (0,1.22)$ has a jump discontinuity at $k=k_c=0.09653$.
However, in principle, we should be getting $$I(0\le k,<k_c)>29/11~~~~~~(5)$$ Against the result (3), I managed to re-evaluate (1) for $k<k_c$ as $$I(k)=\int_{0}^{\epsilon(\nu)} f(x) dx+ \int_{\epsilon(k)}^{1} f(x) dx,~~ \epsilon(k)=10^{-n}, n \in N.$$ Using NIntegrate again, we have achieved: $I(0.003)=3$, when $n=98$; $I(0.002)=3$ when $n=148$; $I(0.0015)=3$ when $n=198$ and it is difficult to get further. By doing this we have pushed the critical value of $k_c$ from $0.09653$ to $0.0015$ to justify the time more than that (5) of the famous cycloid.
We know that (1) for very small values of $k$ becomes cumbersome as the integrand $f(x)$ takes large values in extremely small domain of extremely small values of $x \sim \epsilon(k)$
I have two questions
(1): Is there a better numerical package for finding the integrals like (1) for very small values of $k<0.1$?
(2) Can there be a way to prove analytically that $I(k) \sim 3$ for $0<k<1/10$?
Maybe $I(k)$ is discontinuous at $k=0$ such that $I(0)=1, I(0^+)=3$.