∫ 0 1 ( 3 + x x ) d x = ∫ 0 1 ( 3 + x 1 x 1 2 ) d x = ∫ 0 1 ( 3 + x 1 + 1 2 ) d x = ∫ 0 1 ( 3 + x 3 2 ) d x = 3 x + x 3 2 + 1 3 2 + 1 = 3 x + x 5 2 5 2 = 3 x + 2 x 5 / 2 5 {\displaystyle {\begin{aligned}\int _{0}^{1}\left(3+x{\sqrt {x}}\right)dx&=\int _{0}^{1}\left(3+x^{1}{x}^{\frac {1}{2}}\right)dx=\int _{0}^{1}\left(3+x^{1+{\frac {1}{2}}}\right)dx=\int _{0}^{1}\left(3+x^{\frac {3}{2}}\right)dx\\&=3x+{\frac {x^{{\frac {3}{2}}+1}}{{\frac {3}{2}}+1}}=3x+{\frac {x^{\frac {5}{2}}}{\frac {5}{2}}}=3x+{\frac {2x^{5/2}}{5}}\end{aligned}}}
