<s>[INST] Show that $\frac{1}{3}<\int_0^1\frac{1}{1+x+x^2} \, dx <\frac{\pi}{4}$ Show that $$\frac{1}{3}<\int_0^1\frac{1}{1+x+x^2}\,dx <\frac{\pi}{4}$$
I want to use if $f<g<h$ then $\int f<\int g<\int h$ formula for Riemann integration.
$1+x^2<1+x+x^2$ and it will give RHS as $$\frac{1}{1+x+x^2}<\frac{1}{1+x^2}$$ 
How to choose function $f$ and $h.$
 [/INST] Alternative approach: for any $x\in(0,1)$ we have 
$$ \frac{1}{1+x+x^2}=\frac{1-x}{1-x^3}=(1-x)\sum_{n\geq 0}x^{3n}=\sum_{n\geq 0}x^{3n}-x^{3n+1} \tag{1}$$
hence
$$ \mathcal{J}=\int_{0}^{1}\frac{dx}{1+x+x^2} = \sum_{n\geq 0}\frac{1}{(3n+1)(3n+2)} \tag{2}$$
and the series in the RHS of $(2)$ can be approximated through the Hermite-Hadamard inequality, since $g(x)=\frac{1}{(3x+1)(3x+2)}$ is convex on $\mathbb{R}^+$. We get
$$ \frac{1}{2}+\frac{1}{20}+\int_{2}^{+\infty}g(x)\,dx\leq \mathcal{J}\leq \frac{1}{2}+\frac{1}{20}+\int_{3/2}^{+\infty}g(x)\,dx \tag{3}$$
$$\frac{11}{20}+\frac{1}{3}\log\left(1+\frac{1}{7}\right)\leq \mathcal{J}\leq \frac{11}{20}+\frac{1}{3}\log\left(1+\frac{2}{11}\right)\tag{4}$$
so the distance between $\mathcal{J}$ and $\frac{3}{5}$ is at most $\frac{1}{100}$.
(2) and the reflection formula for the $\psi$ function imply $\mathcal{J}=\frac{\pi}{3\sqrt{3}}$, which can be proved by just completing the square, too. Another interesting approach is creative telescoping:
$$\begin{eqnarray*}\sum_{n\geq 0}\frac{1}{(3n+1)(3n+2)}&=&\frac{1}{2}+\sum_{n\geq 0}\left(\frac{1}{9(n+1)}-\frac{1}{9(n+2)}-\frac{2}{(3n+3)(3n+4)(3n+5)(3n+6)}\right)\\&=&\frac{1}{2}+\frac{1}{9}-\frac{1}{180}-2\sum_{n\geq 0}\frac{1}{(3n+6)(3n+7)(3n+8)(3n+9)}\end{eqnarray*} $$
also leading to the accelerated series $\mathcal{J}=\sum_{n\geq 1}\frac{1}{n\binom{2n}{n}}$.
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