Question

Найти интеграл функции

Answer 1

$\int \sqrt{\dfrac{x+9}{x}}\, dx=\Big[\; t^2=\dfrac{x+9}{x}\; ,\; t^2x=x+9\; ,\; x(t^2-1)=9\; ,\; x=\dfrac{9}{t^2-1}\; ,\\\\\\dx=-\dfrac{18t}{(t^2-1)^2}\, dt\; \Big]=\int \dfrac{-18\cdot t^2}{(t^2-1)}\, dt=-18\int \Big(1+\dfrac{1}{t^2-1}\Big)\, dt=\\\\\\=-18\cdot \Big(t+\dfrac{1}{2}\cdot ln\Big|\dfrac{t-1}{t+1}\Big|\Big)+C=-18\cdot \sqrt{\dfrac{x+9}{x}}-9\cdot ln\Big|\dfrac{\sqrt{x+9}-\sqrt{x}}{\sqrt{x++9}+\sqrt{x}}\Big|+C$