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

1. $$\int_{1}^{2}\frac{5x-2}{\sqrt[3]{x}}\,dx$$
2. $$\int_{1}^{3}\frac{3x-1}{\sqrt{x}}\,dx$$
3. $$\int_{2}^{7}\frac{4}{\sqrt{x+2}}\,dx$$

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

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

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

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

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

   $$=6\sqrt{3}-2\sqrt{3}-2+2=4\sqrt{3}.$$

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

   $$=8\sqrt{x+2}\Bigg|_{2}^{7}$$

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

Ответ: 1) $$3\sqrt[3]{4}$$; 2) $$4\sqrt{3}$$; 3) $$8$$.