The question I have been given is
Prove that $$\frac{1}{2}<\int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1-x^{2n}}} dx\leq 0.52359$$ for any integer $n\geq 1$, and that $$\frac{1}{2}<\int_{0}^{1} \frac{1}{\sqrt{4-x+x^3}} dx\leq 0.52359$$
I can show that it works for $n=1,2$ but I don't know how to approach proving this for all $n\geq 1$. I also can't see how the two equations relate. I have asked my professor and he did not know what to do.
If there is any direction you could point me in or any part of the proof you could show I would really appreciate it.
Thank you!
For the first integral:
That $0.52359$ constant is supposed to be $\frac \pi6$. Thats the integral when $n=1$. The integrand is decreasing on $(0,\frac 12)$, and it's graph forms something very close to a rectangle with the lines $x=0$ and $x=1$. This can be seen since $\frac{1}{1-(0)^{2n}}=1$ regardless the value of $n$, and $\frac{1}{1-(\frac 12)^{2n}}$ decreases from $\frac 43$ to just over $1$ as $n$ increases, and the function is continuous in the interim.