Question Number 141530 by sarkor last updated on 20/May/21
Commented by mohammad17 last updated on 20/May/21
$$=\int\:\frac{{x}^{\mathrm{2}} −\mathrm{9}+\mathrm{9}}{\:\sqrt{{x}−\mathrm{3}}}{dx}=\int\:\frac{\left({x}−\mathrm{3}\right)\left({x}+\mathrm{3}\right)}{\:\sqrt{{x}−\mathrm{3}}}{dx}+\int\:\frac{\mathrm{9}}{\:\sqrt{{x}−\mathrm{3}}}{dx} \\ $$$$ \\ $$$$=\int\left(\sqrt{{x}−\mathrm{3}}\right)\left({x}+\mathrm{3}\right){dx}+\int\mathrm{9}\left({x}−\mathrm{3}\right)^{−\frac{\mathrm{1}}{\mathrm{2}}} {dx} \\ $$$$ \\ $$$$=\int\left(\sqrt{{x}−\mathrm{3}}\right)\left({x}−\mathrm{3}+\mathrm{6}\right){dx}+\mathrm{9}\int\left({x}−\mathrm{3}\right)^{−\frac{\mathrm{1}}{\mathrm{2}}} {dx} \\ $$$$ \\ $$$$=\int\left({x}−\mathrm{3}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} {dx}+\mathrm{6}\int\:\left({x}−\mathrm{3}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} {dx}+\mathrm{9}\int\:\left({x}−\mathrm{3}\right)^{−\frac{\mathrm{1}}{\mathrm{2}}} {dx} \\ $$$$ \\ $$$$=\frac{\mathrm{2}}{\mathrm{5}}\sqrt{\left({x}−\mathrm{3}\right)^{\mathrm{5}} }+\mathrm{4}\sqrt{\left({x}−\mathrm{3}\right)^{\mathrm{3}} }+\mathrm{18}\sqrt{\left({x}−\mathrm{3}\right)}+{C} \\ $$$$ \\ $$$${by}::\:\langle{m}.{o}\rangle \\ $$