Question Number 20292 by tammi last updated on 25/Aug/17
$$\int\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} +\sqrt{{x}−\mathrm{1}}} \\ $$
Answered by $@ty@m last updated on 25/Aug/17
$$=\int\frac{{dx}}{\:\sqrt{{x}+\mathrm{1}}+\sqrt{{x}−\mathrm{1}}} \\ $$$$=\int\frac{\sqrt{{x}+\mathrm{1}}−\sqrt{{x}−\mathrm{1}}}{\left({x}+\mathrm{1}\right)−\left({x}−\mathrm{1}\right)}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\sqrt{{x}+\mathrm{1}}{dx}−\frac{\mathrm{1}}{\mathrm{2}}\int\sqrt{{x}−\mathrm{1}}{dx} \\ $$$$=\frac{\left({x}+\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }{\mathrm{3}}−\frac{\left({x}−\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }{\mathrm{3}}+{C} \\ $$