13 ∫(√x+1)(x-1/∛x) dx

Краткое пояснение:

Разбираемся:

\(\int (\sqrt{x} + 1)(x - \frac{1}{\sqrt[3]{x}}) dx = \int (x^{\frac{1}{2}} + 1)(x - x^{-\frac{1}{3}}) dx = \int (x^{\frac{3}{2}} - x^{\frac{1}{2} - \frac{1}{3}} + x - x^{-\frac{1}{3}}) dx = \)

\(= \int (x^{\frac{3}{2}} - x^{\frac{1}{6}} + x - x^{-\frac{1}{3}}) dx = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} - \frac{x^{\frac{7}{6}}}{\frac{7}{6}} + \frac{x^2}{2} - \frac{x^{\frac{2}{3}}}{\frac{2}{3}} + C = \frac{2}{5}x^{\frac{5}{2}} - \frac{6}{7}x^{\frac{7}{6}} + \frac{1}{2}x^2 - \frac{3}{2}x^{\frac{2}{3}} + C\)

Ответ:

\(\frac{2}{5}x^{\frac{5}{2}} - \frac{6}{7}x^{\frac{7}{6}} + \frac{1}{2}x^2 - \frac{3}{2}x^{\frac{2}{3}} + C\)