Question

Solve : $0<{\int }_0^1\frac{x^{7}dx}{(1+x^8{)}^{\frac{1}{3}}}<\frac{1}{8}$

Answer 1

$\int \frac{x^7}{\sqrt[3]{31+x^8}}dx$

$u=1+x^8\to du=8x^{7}dx$

$=\frac{1}{8}\int \frac{1}{\sqrt[3]{3u}}du$

$=\frac{1}{8}\frac{3u^{\frac{2}{3}}}{16}$

$=\frac{1}{8}\frac{3(1+x^8{)}^{\frac{2}{3}}}{16}+C$

${\int }_0^1\frac{x^7}{\sqrt[3]{31+x^8}}dx$

$\frac{3}{2^{\frac{10}{3}}}-\frac{3}{16}$

$=0.110$