Question Number 20293 by tammi last updated on 25/Aug/17
$$\int\sqrt{\frac{{a}+{x}}{{x}}{dx}} \\ $$
Answered by $@ty@m last updated on 25/Aug/17
$$=\int\frac{{a}+{x}}{\:\sqrt{{x}\left({a}+{x}\right)}}{dx} \\ $$$$=\int\frac{{a}+{x}}{\:\sqrt{{ax}+{x}^{\mathrm{2}} }}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{2}{x}+{a}+{a}}{\:\sqrt{{ax}+{x}^{\mathrm{2}} }}{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{2}{x}+{a}}{\:\sqrt{{ax}+{x}^{\mathrm{2}} }}{dx}+\frac{{a}}{\mathrm{2}}\int\frac{{dx}}{\:\sqrt{{ax}+{x}^{\mathrm{2}} }} \\ $$$$=\sqrt{{ax}+{x}^{\mathrm{2}} }+\frac{{a}}{\mathrm{2}}\int\frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}.\frac{{a}}{\mathrm{2}}{x}+\left(\frac{{a}}{\mathrm{2}}\right)^{\mathrm{2}} −\left(\frac{{a}}{\mathrm{2}}\right)^{\mathrm{2}} }}{dx} \\ $$$$=\sqrt{{ax}+{x}^{\mathrm{2}} }+\frac{{a}}{\mathrm{2}}\int\frac{{dx}}{\:\sqrt{\left({x}+\frac{{a}}{\mathrm{2}}\right)^{\mathrm{2}} −\left(\frac{{a}}{\mathrm{2}}\right)^{\mathrm{2}} }} \\ $$$$=\sqrt{{ax}+{x}^{\mathrm{2}} }+\frac{{a}}{\mathrm{2}}{ln}\mid{x}+\frac{{a}}{\mathrm{2}}+\sqrt{{ax}+{x}^{\mathrm{2}} }\mid+{C} \\ $$