Question

Salut, am si eu nevoie de ajutor la ex. 998( explicatii pas cu pas, va rog )

Answer 1

$\displaystyle I = \int_{0}^1\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\,dx=\\ \\=\int_{0}^1 \dfrac{1-x}{\sqrt{(1-x)(1+x)}} \, dx= \int_{0}^1 \dfrac{1-x}{\sqrt{1-x^2}}\, dx=\\\\ =\int_{0}^1 \dfrac{1}{\sqrt{1-x^2}}\, dx+ \int_{0}^1 \dfrac{-x}{\sqrt{1-x^2}}\, dx =\\ \\=\arcsin(x)\Big|_{0}^1+\int_{0}^1\Big(\sqrt{1-x^2}\Big)'\, dx =\\ \\ = \dfrac{\pi}{2}+ \Big(\sqrt{1-x^2}\Big)\Big|_0^1 = \boxed{\dfrac{\pi}{2}-1}$