Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 1009

\[\boxed{\mathbf{1009}\mathbf{.}}\]

\[1)\int_{1}^{2}{\frac{5x - 2}{\sqrt[3]{x}}\text{\ dx}} =\]

\[= \int_{1}^{2}{\frac{5x - 2}{x^{\frac{1}{3}}}\text{\ dx}} =\]

\[= \int_{1}^{2}{\left( 5x^{\frac{2}{3}} - 2x^{- \frac{1}{3}} \right)\text{\ dx}} =\]

\[= \left. \ \left( 5 \bullet x^{\frac{5}{3}}\ :\frac{5}{3} - 2 \bullet x^{\frac{2}{3}}\ :\frac{2}{3} \right) \right|_{1}^{2} =\]

\[= \left. \ \left( 3 \bullet x\sqrt[3]{x^{2}} - 3 \bullet \sqrt[3]{x^{2}} \right) \right|_{1}^{2} =\]

\[= 6 \bullet \sqrt[3]{4} - 3 \bullet \sqrt[3]{4} - 3 + 3 = 3\sqrt[3]{4};\]

\[2)\int_{1}^{3}{\frac{3x - 1}{\sqrt{x}}\text{\ dx}} =\]

\[= \int_{1}^{3}{\frac{3x - 1}{x^{\frac{1}{2}}}\text{\ dx}} =\]

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

\[= \left. \ \left( 3 \bullet x^{\frac{3}{2}}\ :\frac{3}{2} - x^{\frac{1}{2}}\ :\frac{1}{2} \right) \right|_{1}^{3} =\]

\[= \left. \ \left( 2x\sqrt{x} - 2\sqrt{x} \right) \right|_{1}^{3} =\]

\[= 6\sqrt{3} - 2\sqrt{3} - 2 + 2 = 4\sqrt{3};\]

\[3)\int_{2}^{7}{\frac{4}{\sqrt{x + 2}}\text{\ dx}} =\]

\[= \int_{2}^{7}{4(x + 2)^{- \frac{1}{2}}\text{\ dx}} =\]

\[= \left. \ 4 \bullet (x + 2)^{\frac{1}{2}}\ :\frac{1}{2}\ \right|_{2}^{7} =\]

\[= \left. \ 8\sqrt{x + 2}\ \right|_{2}^{7} =\]

\[= 8 \bullet \sqrt{7 + 2} - 8 \bullet \sqrt{2 + 2} =\]

\[= 8\sqrt{9} - 8\sqrt{4} = 8 \bullet 3 - 8 \bullet 2 = 8.\]