<s>[INST] question on integrals Let $\displaystyle A=\int_0^1 \frac{dx}{1+x^8}$. Then which of the following are true:
1) $A\lt 1$,                                                                                
2) $A\gt 1$,                                                                                    
3) $A\lt \frac{\pi}{4}$,                                                                      
4) $A\gt\frac{\pi}{4}$.
 [/INST] If $0\lt x\lt 1$, we have
$$\frac{1}{1+x^2}\lt \frac{1}{1+x^8}\lt \frac{1}{1+0^2}.$$
Hence, we have
$$\frac{\pi}{4}=\int_0^1\frac{1}{1+x^2}dx\lt\int_0^1\frac{1}{1+x^8}dx\lt\int_0^1\frac{1}{1+0^2}dx=1.$$
That leads that the answer $4)$ and $1)$.
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