Question

помогите вычислить определенный интеграл
$\int\limits^4_0 {\frac{\sqrt{x}}{4+x}} \, dx$

Answer 1

$\int\limits^4_0 {\frac{\sqrt{x}}{x+4}} \, dx =\\\\ t=\sqrt{x}\\\\ dt=\frac{1}{2\sqrt{x}}dx\\ 2\sqrt{x}dt=dx\\ dx=2tdt\\\\ = \int\limits^{\sqrt{4}}_{\sqrt {0}} {\frac{t}{t^2+4}} \, 2tdt= 2*\int\limits^2_0 {\frac{t^2}{t^2+4}} \, dt =2* \int\limits^2_0 {\frac{t^2+4-4}{t^2+4}} \, dt =\\\\ =2* \int\limits^2_0 {(1-\frac{4}{t^2+4})} \, dt =2* [\int\limits^2_0 {1}\, dt -4*\int\limits^2_0 {\frac{1}{t^2+2^2}} \, dt]=\\\\ =2* [t|^2_0 -4*\frac{1}{2}*arctg(\frac{t}{2})|^2_0]=\\\\ =2*[2-2*(arctg(1)-arctg(0))]=$

$=2*[2-2*(\frac{\pi}{4}-0)]=4-\pi$