Question Number 95691 by mathmax by abdo last updated on 27/May/20
$$\mathrm{calculate}\:\int\:\:\frac{\mathrm{x}+\mathrm{1}}{\:\sqrt{\left(\mathrm{x}+\mathrm{3}\right)\left(\mathrm{2}−\mathrm{x}\right)}}\mathrm{dx} \\ $$
Commented by Tony Lin last updated on 27/May/20
$$\int\frac{{x}+\mathrm{1}}{\:\sqrt{\left({x}+\mathrm{3}\right)\left(\mathrm{2}−{x}\right)}}{dx} \\ $$$$=\int\frac{{x}+\mathrm{1}}{\:\sqrt{−{x}^{\mathrm{2}} −{x}+\mathrm{6}}}{dx} \\ $$$$=\int\frac{{x}+\frac{\mathrm{1}}{\mathrm{2}}}{\:\sqrt{−{x}^{\mathrm{2}} −{x}+\mathrm{6}}}{dx}+\int\frac{\frac{\mathrm{1}}{\mathrm{2}}}{\:\sqrt{−{x}^{\mathrm{2}} −{x}+\mathrm{6}}}{dx} \\ $$$$=−\sqrt{−{x}^{\mathrm{2}} −{x}+\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{1}}{\:\sqrt{\left(\frac{\mathrm{5}}{\mathrm{2}}\right)^{\mathrm{2}} −\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }}{dx} \\ $$$$=−\sqrt{−{x}^{\mathrm{2}} −{x}+\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{2}}{sin}^{−\mathrm{1}} \frac{{x}−\frac{\mathrm{1}}{\mathrm{2}}}{\frac{\mathrm{5}}{\mathrm{2}}}+{c} \\ $$$$=−\sqrt{−{x}^{\mathrm{2}} −{x}+\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{2}}{sin}^{−\mathrm{1}} \frac{\mathrm{2}{x}−\mathrm{1}}{\mathrm{5}}+{c} \\ $$