exo7 4265

\[\begin{array}{ll} \frac{x+1}{\sqrt{x^2-3x+2}} & \sqrt{x^2-3x+2} + \frac52 \ln\bigl| 2x-3 + 2\sqrt{x^2-3x+2} \bigr| \cr \frac{4x-3}{\sqrt{-4x^2+12x-5}} & -\sqrt{-4x^2+12x-5} + \frac32 \Arcsin(x-3/2) \cr \frac1{2x-x^2+\sqrt{2x-x^2}} & \frac{1-\sqrt{2x-x^2}}{x-1} \cr \frac1{2+\sqrt{1+x}+\sqrt{3-x}} & \sqrt{1+x} - \sqrt{3-x} - \Arcsin\Bigl(\frac{x-1}2 \Bigr) \quad(\text{poser } x = 1+2\cos\varphi) \cr \frac{2+\sqrt{x+3}}{1+\sqrt{x+4}} & (\sqrt{x+3}+4)(\sqrt{x+4}-2) -4\ln(1+\sqrt{x+4}\,) +\ln(\sqrt{x+3}+\sqrt{x+4}\,) \cr x+\sqrt{a^2+x^2} & \frac{(x+\sqrt{a^2+x^2}\,)^2}4 + \frac{a^2}2 \ln( x+ \sqrt{a^2+x^2}\, ) \cr (x+\sqrt{a^2+x^2}\,)^n & \frac{(x+\sqrt{a^2+x^2}\,)^{n+1}}{2(n+1)} + a^2\frac{(x+\sqrt{a^2+x^2}\,)^{n-1}}{2(n-1)} \quad (n\ne1) \cr \frac1{\sqrt[3]{1+x^3}} & \frac16 \ln\Bigl[ \frac{u^2+u+1}{(u-1)^2} \Bigr] - \frac1{\sqrt3} \Arctan \frac{2u+1}{\sqrt3} , u = \sqrt[3]{1+1/x^3}\quad (\text{poser } v=1/x^3) \cr \end{array}\]