<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] A better upper bound: by the convexity of the integrand in $[0,1]$:
$$\forall x\in[0,1]:\qquad\frac1{1 + x + x^2}\le 1 - \frac{2x}3,$$ 
and this implies
$$\int_0^1\frac{1}{1+x+x^2}dx\le\int_0^1(1 - 2x/3)\,dx = \frac23.$$
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