Question Number 41845 by maxmathsup by imad last updated on 13/Aug/18
$$\left.\mathrm{1}\right){find}\:\:\:\:\int\:\:\:\:\:\:\:\:\:\frac{{x}}{\:\sqrt{\mathrm{1}+{x}}\:+\sqrt{\mathrm{1}−{x}}}\:{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\:\:\:\frac{{x}}{\:\sqrt{\mathrm{1}+{x}}\:+\sqrt{\mathrm{1}−{x}}}\:{dx} \\ $$
Answered by MJS last updated on 13/Aug/18
$$\int\frac{{x}}{\:\sqrt{\mathrm{1}+{x}}+\sqrt{\mathrm{1}−{x}}}{dx}=\frac{\mathrm{1}}{\mathrm{2}}\int\sqrt{\mathrm{1}+{x}}{dx}−\frac{\mathrm{1}}{\mathrm{2}}\int\sqrt{\mathrm{1}−{x}}{dx}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}\left(\sqrt{\left(\mathrm{1}+{x}\right)^{\mathrm{3}} }+\sqrt{\left(\mathrm{1}−{x}\right)^{\mathrm{3}} }\right)+{C} \\ $$