Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 367

\[1)\ \int_{0}^{3}{x^{2}\text{\ dx}} = \left. \ \frac{x^{3}}{3} \right|_{0}^{3} = \frac{3^{3}}{3} = 3^{2} = 9;\]

\[2)\ \int_{- 2}^{3}{2x\ dx} = \left. \ x^{2} \right|_{- 2}^{3} =\]

\[= 3^{2} - ( - 2)^{2} = 9 - 4 = 5;\]

\[3)\ \int_{1}^{2}{\frac{1}{x^{3}}\text{\ dx}} = \left. \ \frac{x^{- 2}}{- 2} \right|_{1}^{2} =\]

\[= \frac{2^{- 2}}{- 2} - \frac{1^{- 2}}{- 2} = - \frac{1}{2 \bullet 4} + \frac{1}{2} =\]

\[= \frac{4}{8} - \frac{1}{8} = \frac{3}{8};\]

\[4)\ \int_{4}^{9}{\frac{1}{\sqrt{x}}\text{\ dx}} = \left. \ 2x^{\frac{1}{2}} \right|_{4}^{9} =\]

\[= 2 \bullet 9^{\frac{1}{2}} - 2 \bullet 4^{\frac{1}{2}} = 2 \bullet 3 - 2 \bullet 2 = 2.\]