Question

Salut, am si eu nevoie de ajutor la ex.
937 (explicatii pas cu pas).

Answer 1

$\displaystyle I = \int \dfrac{1}{\sqrt{x(1+x)}}\, dx =\displaystyle 2\int \dfrac{\sqrt{1+x}}{\sqrt{1+x}\cdot \sqrt{x}\cdot 2\sqrt{1+x}}\, dx\\ \\\\ \sqrt{1+x}= t \Rightarrow \dfrac{1}{2\sqrt{1+x}}\, dx = dt \\ \\ 1+x = t^2 \Rightarrow x = t^2-1 \\ \\ \\ I =2 \int \dfrac{t}{t\sqrt{t^2-1}}\, dt = 2\int \dfrac{1}{\sqrt{t^2-1}}\, dt = 2\ln(t+\sqrt{t^2-1})+C = \\ \\ = 2\ln(\sqrt{1+x}+\sqrt{1+x-1}) +C = \boxed{2\ln(\sqrt{x}+\sqrt{1+x})+C}$