5.4 Indefinite Integrals and the Net Change Theorem/31

${\begin{aligned}\int _{0}^{1}x\left({\sqrt[{3}]{x}}+{\sqrt[{4}]{x}}\right)dx&=\int _{0}^{1}x\left(x^{\frac {1}{3}}+x^{\frac {1}{4}}\right)dx=\int _{0}^{1}\left(x^{\frac {4}{3}}+x^{\frac {5}{4}}\right)dx\\[2ex]&=\left({\frac {3x^{\frac {7}{3}}}{7}}+{\frac {4x^{\frac {9}{4}}}{9}}\right){\Bigg |}_{0}^{1}\\[2ex]&={\frac {3}{7}}+{\frac {4}{9}}={\frac {27+28}{7\cdot 9}}={\frac {55}{63}}\end{aligned}}$