Упр.1009 ГДЗ Алимов 10-11 класс (Алгебра)

2) интеграл (1;3) (3x-1)/ корень x dx;
3) интеграл (2;7) (4/ корень (x+2) dx.

1. $$\int_{1}^{2}\frac{5x-2}{\sqrt[3]{x}}\,dx=\int_{1}^{2}\left(5x^{\frac23}-2x^{-\frac13}\right)\,dx$$

   $$=\left(5\cdot \frac{x^{\frac53}}{\frac53}-2\cdot \frac{x^{\frac23}}{\frac23}\right)\Bigg|_{1}^{2} =\left(3x^{\frac53}-3x^{\frac23}\right)\Bigg|_{1}^{2}$$

   $$=3\cdot 2^{\frac53}-3\cdot 2^{\frac23}-3+3 =3\sqrt[3]{4}.$$

2. $$\int_{1}^{3}\frac{3x-1}{\sqrt{x}}\,dx=\int_{1}^{3}\left(3x^{\frac12}-x^{-\frac12}\right)\,dx$$

   $$=\left(3\cdot \frac{x^{\frac32}}{\frac32}-\frac{x^{\frac12}}{\frac12}\right)\Bigg|_{1}^{3} =\left(2x\sqrt{x}-2\sqrt{x}\right)\Bigg|_{1}^{3}$$

   $$=2\cdot 3\sqrt{3}-2\sqrt{3}-2\cdot 1+2\cdot 1=4\sqrt{3}.$$

3. $$\int_{2}^{7}\frac{4}{\sqrt{x+2}}\,dx=\int_{2}^{7}4(x+2)^{-\frac12}\,dx$$

   $$=4\cdot \frac{(x+2)^{\frac12}}{\frac12}\Bigg|_{2}^{7} =8\sqrt{x+2}\Bigg|_{2}^{7}$$

   $$=8\sqrt{9}-8\sqrt{4}=24-16=8.$$

Ответ

$$3\sqrt[3]{4};\ 4\sqrt{3};\ 8$$