Question Number 73483 by abdomathmax last updated on 13/Nov/19
$${find}\:\int\:\:\:\:\frac{{dx}}{{x}+\mathrm{2}−\sqrt{{x}^{\mathrm{2}} −{x}\:+\mathrm{7}}} \\ $$
Answered by MJS last updated on 13/Nov/19
$$\int\frac{{dx}}{{x}+\mathrm{2}−\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{7}}}=\int\frac{{dx}}{{x}+\mathrm{2}−\sqrt{\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\frac{\mathrm{27}}{\mathrm{4}}}}= \\ $$$$\:\:\:\:\:\left[{t}=\frac{\sqrt{\mathrm{3}}}{\mathrm{9}}\left(\mathrm{2}{x}−\mathrm{1}\right)\:\rightarrow\:{dx}=\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{2}}{dt}\right] \\ $$$$=\int\frac{{dt}}{{t}+\frac{\mathrm{5}\sqrt{\mathrm{3}}}{\mathrm{9}}−\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{sinh}\:\mathrm{ln}\:{u}\:=\frac{{u}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{u}}\:\Rightarrow\:{u}={t}+\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}\:\rightarrow\:{dt}=\frac{{u}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}{u}^{\mathrm{2}} }{du}\right] \\ $$$$=\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{2}}\int\frac{{u}^{\mathrm{2}} +\mathrm{1}}{{u}\left(\mathrm{5}{u}−\mathrm{3}\sqrt{\mathrm{3}}\right)}{du}= \\ $$$$=\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{10}}\int{du}−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{du}}{{u}}+\frac{\mathrm{26}}{\mathrm{25}}\int\frac{{du}}{{u}−\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{5}}}= \\ $$$$=\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{10}}{u}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:{u}\:+\frac{\mathrm{26}}{\mathrm{25}}\mathrm{ln}\:\left(\mathrm{5}{u}−\mathrm{3}\sqrt{\mathrm{3}}\right)\:= \\ $$$$… \\ $$$$=\frac{\mathrm{2}{x}−\mathrm{1}+\mathrm{2}\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{7}}}{\mathrm{10}}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\left(\mathrm{2}{x}−\mathrm{1}+\mathrm{2}\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{7}}\right)\:+\frac{\mathrm{26}}{\mathrm{25}}\mathrm{ln}\:\left(\mathrm{5}{x}−\mathrm{16}+\mathrm{5}\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{7}}\right)\:+{C} \\ $$
Commented by abdomathmax last updated on 17/Nov/19
$${thanks}\:{sir}\:{mjs}. \\ $$