Let \(I=\int_0^1\sqrt[4]{1-x^4}dx, \ J=\int_0^1\sqrt[8]{1-x^8}dx, \text{ and } K=\int_0^1\sqrt[4]{1+x^4}dx \), compare the values of these integrals.
Solution: I<J<1<K.
Notice that
\(\sqrt[4]{1-x^4}<\sqrt[8]{1-x^8} \Leftarrow (1-x^4)^2<1-x^8 \Leftrightarrow -2x^4<0,\text{ which is obviously true for } x\in(0,1).\)