Question

Интеграл от1до2 (x+2/корень из x)dx вычислите

Answer 1

$\int\limits^2_1 {\frac{x+2}{\sqrt{x}}} \, dx = \int\limits^2_1 {(\frac{x}{\sqrt{x}}+\frac{2}{\sqrt{x}})} \, dx = \int\limits^2_1 {\sqrt{x}} \, dx + \int\limits^2_1 {\frac{2}{\sqrt{x}}} \, dx =\\\\=\frac{2x^{\frac{3}{2}}}{3}|_1^2+2\cdot 2\sqrt{x}|_1^2=\frac{2}{3}(2^{\frac{3}{2}}-2)+4(\sqrt2-\sqrt1)=\\\\=\frac{2}{3}(2\sqrt2-2)+4(\sqrt2-1)=\frac{4}{3}(\sqrt2-1)+4\sqrt2-1)=\\\\=\frac{4}{3}\cdot 4(\sqrt2+1)=\frac{16}{3}(\sqrt2+1)$