Question Number 18024 by mondodotto@gmail.com last updated on 13/Jul/17
Answered by sma3l2996 last updated on 14/Jul/17
$${I}=\int\frac{\mathrm{5}{x}+\mathrm{3}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}}{dx}=\frac{\mathrm{5}}{\mathrm{2}}\int\frac{\mathrm{2}{x}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}}{dx}+\int\frac{\mathrm{3}{dx}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}} \\ $$$$=\frac{\mathrm{5}}{\mathrm{2}}\int\frac{\mathrm{2}{x}+\mathrm{4}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}}+\int\frac{\mathrm{3}−\mathrm{10}}{\:\sqrt{\left({x}+\mathrm{2}\right)^{\mathrm{2}} +\mathrm{6}}}{dx} \\ $$$$=\mathrm{5}\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}−\int\frac{\mathrm{7}}{\:\sqrt{\mathrm{6}}\sqrt{\left(\frac{{x}+\mathrm{2}}{\:\sqrt{\mathrm{6}}}\right)^{\mathrm{2}} +\mathrm{1}}}{dx}+{c} \\ $$$${let}\:\:{t}=\frac{{x}+\mathrm{2}}{\:\sqrt{\mathrm{6}}}\Rightarrow{dt}=\frac{{dx}}{\:\sqrt{\mathrm{6}}} \\ $$$$\frac{\mathrm{7}}{\:\sqrt{\mathrm{6}}}\int\frac{{dx}}{\:\sqrt{\left(\frac{{x}+\mathrm{2}}{\:\sqrt{\mathrm{6}}}\right)^{\mathrm{2}} +\mathrm{1}}}=\mathrm{7}\int\frac{{dt}}{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}=\mathrm{7}{ln}\left({t}+\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}\right)+{a} \\ $$$${I}=\mathrm{5}\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}−\mathrm{7}{ln}\left(\frac{{x}+\mathrm{2}}{\:\sqrt{\mathrm{6}}}+\sqrt{\left(\frac{{x}+\mathrm{2}}{\:\sqrt{\mathrm{6}}}\right)^{\mathrm{2}} +\mathrm{1}}\right)+{C}_{\mathrm{1}} \\ $$$$=\mathrm{5}\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}−\mathrm{7}{ln}\left({x}+\mathrm{2}+\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}\right)−\mathrm{7}{ln}\sqrt{\mathrm{6}}+{C}_{\mathrm{1}} \\ $$$${I}=\mathrm{5}\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}−\mathrm{7}{ln}\left({x}+\mathrm{2}+\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{10}}\right)+{C} \\ $$
Commented by mondodotto@gmail.com last updated on 14/Jul/17
$$\mathrm{i}\:\mathrm{preciate}\:\mathrm{sir}\:\mathrm{thanx} \\ $$
Answered by Abbas-Nahi last updated on 14/Jul/17
$$\int\frac{\:\mathrm{5}{x}+\mathrm{3}}{\:\sqrt{\left({x}+\mathrm{2}\right)^{\mathrm{2}} \:+\mathrm{6}\:\:}\:}\:{dx} \\ $$$${let}\:{u}={x}+\mathrm{2}\:\:\:\:{du}={dx} \\ $$$$\int\frac{\mathrm{5}\left({u}−\mathrm{2}\right)+\mathrm{3}}{\:\sqrt{{u}^{\mathrm{2}} +\mathrm{6}}\:}\:{du}=\int\frac{\:\mathrm{5}{u}−\mathrm{7}}{\:\sqrt{{u}^{\mathrm{2}} +\mathrm{6}}\:}\:{du} \\ $$$$\mathrm{5}\int\frac{{u}}{\:\sqrt{{u}^{\mathrm{2}} +\mathrm{6}}\:}\:{du}−\mathrm{7}\int\frac{\mathrm{1}}{\:\sqrt{{u}^{\mathrm{2}} +\mathrm{6}}\:}{du} \\ $$$$\frac{\mathrm{5}}{\mathrm{2}}\int\frac{\mathrm{2}{u}}{\:\sqrt{{u}^{\mathrm{2}} +\mathrm{6}}\:}\:{du}−\frac{\mathrm{7}}{\:\sqrt{\mathrm{6}}\:}\:\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\left(\frac{{u}}{\:\sqrt{\mathrm{6}}}\:\right)^{\mathrm{2}\:} \:\:\:}}\:{du} \\ $$$${then}\:{complete}\:{the}\:{solution}…. \\ $$
Commented by mondodotto@gmail.com last updated on 14/Jul/17
$$\mathrm{waoohh}\:\mathrm{nice1}\:\mathrm{thanx} \\ $$