Question Number 101531 by mhmd last updated on 03/Jul/20
$${find}\:\int\sqrt{{ax}−{x}^{\mathrm{2}} }{dx} \\ $$
Answered by Mr.D.N. last updated on 03/Jul/20
$$\:\:\mathrm{I}\:=\:\int\sqrt{\mathrm{ax}−\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$$$\:\:\:=\:\int\sqrt{−\left(\mathrm{x}^{\mathrm{2}} −\mathrm{ax}\right)}\:\:\:\mathrm{dx} \\ $$$$\:\:=\:\:\int\:\:\sqrt{\left(\frac{\mathrm{a}}{\mathrm{2}}\right)^{\mathrm{2}} −\left(\mathrm{x}−\frac{\mathrm{a}}{\mathrm{2}}\right)^{\mathrm{2}} }\:\:\:\:\:\mathrm{dx} \\ $$$$\:\:=\:\:\frac{\left(\mathrm{x}−\frac{\mathrm{a}}{\mathrm{2}}\right)\sqrt{\left(\frac{\mathrm{a}}{\mathrm{2}}\right)^{\mathrm{2}} −\left(\mathrm{x}−\frac{\mathrm{a}}{\mathrm{2}}\right)^{\mathrm{2}} }}{\mathrm{2}}\:+\:\frac{\left(\frac{\mathrm{a}}{\mathrm{2}}\right)^{\mathrm{2}} }{\mathrm{2}}\mathrm{sin}^{−\mathrm{1}} \frac{\left(\mathrm{x}−\frac{\mathrm{a}}{\mathrm{2}}\right)}{\frac{\mathrm{a}}{\mathrm{2}}}+\mathrm{C} \\ $$$$=\:\frac{\frac{\mathrm{2x}−\mathrm{a}}{\mathrm{2}}\sqrt{\mathrm{ax}−\mathrm{x}^{\mathrm{2}} }}{\mathrm{2}}\:+\:\frac{\mathrm{a}^{\mathrm{2}} }{\mathrm{8}}\:\mathrm{sin}^{−\mathrm{1}} \frac{\mathrm{2x}−\mathrm{a}}{\mathrm{a}}+\mathrm{C} \\ $$$$=\:\frac{\left(\mathrm{2x}−\mathrm{a}\right)\sqrt{\mathrm{ax}−\mathrm{x}^{\mathrm{2}} }}{\mathrm{4}}\:+\:\frac{\mathrm{a}^{\mathrm{2}} }{\mathrm{8}}\:\mathrm{sin}^{−\mathrm{1}} \:\frac{\left(\mathrm{2x}−\mathrm{a}\right)}{\mathrm{a}}\:+\mathrm{C}//. \\ $$
Commented by mhmd last updated on 03/Jul/20
$${thank}\:{you}\:{sir} \\ $$