<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] Note that for $x\ge0$, $1+x\le x^2+x+1\le(x+1)^2$.  
Hence, we have
$$\frac13<\frac12=\int_0^1 \frac{1}{(1+x)^2}\,dx\le \int_0^1 \frac{1}{x^2+x+1}\,dx\le \int_0^1 \frac{1}{1+x}\,dx=\log(2)<\frac\pi4$$
which provide tighter bounds then those that were requested.
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