Question Number 73478 by abdomathmax last updated on 13/Nov/19
$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{x}^{\mathrm{3}} −\mathrm{3}}{\:\sqrt{{x}^{\mathrm{2}} −{x}\:+\mathrm{2}}}{dx} \\ $$
Answered by MJS last updated on 13/Nov/19
$$\int\frac{{x}^{\mathrm{3}} −\mathrm{3}}{\:\sqrt{{x}^{\mathrm{3}} −{x}+\mathrm{2}}}{dx}=\int\frac{{x}^{\mathrm{3}} −\mathrm{3}}{\:\sqrt{\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\frac{\mathrm{7}}{\mathrm{4}}}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{\mathrm{2}{x}−\mathrm{1}}{\:\sqrt{\mathrm{7}}}\:\rightarrow\:{dx}=\frac{\sqrt{\mathrm{7}}}{\mathrm{2}}{dt}\right] \\ $$$$=\frac{\mathrm{1}}{\mathrm{8}}\int\frac{\mathrm{7}\sqrt{\mathrm{7}}{t}^{\mathrm{3}} +\mathrm{21}{t}^{\mathrm{2}} +\mathrm{3}\sqrt{\mathrm{7}}{t}−\mathrm{23}}{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}{dt}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{sinh}\:\mathrm{ln}\:{u}\:=\frac{{u}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{u}}\:\Rightarrow\:{u}={t}+\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}\:\rightarrow\:{dt}=\frac{\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}{{t}+\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}{du}=\frac{{u}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}{u}^{\mathrm{2}} }{du}\right] \\ $$$$=\frac{\mathrm{1}}{\mathrm{64}}\int\frac{\mathrm{7}\sqrt{\mathrm{7}}{u}^{\mathrm{6}} +\mathrm{42}{u}^{\mathrm{5}} −\mathrm{9}\sqrt{\mathrm{7}}{u}^{\mathrm{4}} −\mathrm{268}{u}^{\mathrm{3}} +\mathrm{9}\sqrt{\mathrm{7}}{u}^{\mathrm{2}} +\mathrm{42}{u}−\mathrm{7}\sqrt{\mathrm{7}}}{{u}^{\mathrm{4}} }{du} \\ $$$$\mathrm{and}\:\mathrm{it}'\mathrm{s}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{this} \\ $$
Commented by abdomathmax last updated on 17/Nov/19
$${thank}\:{you}\:{sir}. \\ $$