∫ 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 {\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}}\end{aligned}}}
