Question Number 123314 by aurpeyz last updated on 24/Nov/20
$$\int\frac{\mathrm{8}}{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{5}}{dx} \\ $$
Answered by mathmax by abdo last updated on 24/Nov/20
$$\mathrm{I}\:=\mathrm{8}\:\int\:\:\frac{\mathrm{dx}}{\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} \:+\frac{\mathrm{3}}{\mathrm{2}}\mathrm{x}+\frac{\mathrm{5}}{\mathrm{2}}\right)}=\mathrm{4}\int\:\:\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{2}.\frac{\mathrm{3}}{\mathrm{4}}\mathrm{x}+\frac{\mathrm{9}}{\mathrm{16}}+\frac{\mathrm{5}}{\mathrm{2}}−\frac{\mathrm{9}}{\mathrm{16}}} \\ $$$$=\mathrm{4}\int\:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}+\frac{\mathrm{3}}{\mathrm{4}}\right)^{\mathrm{2}} +\frac{\mathrm{31}}{\mathrm{16}}}\:=_{\mathrm{x}+\frac{\mathrm{3}}{\mathrm{4}}=\frac{\sqrt{\mathrm{31}}}{\mathrm{4}}\mathrm{u}} \:\:\:\mathrm{4}×\frac{\mathrm{16}}{\mathrm{31}}\int\:\:\frac{\mathrm{1}}{\mathrm{u}^{\mathrm{2}} \:+\mathrm{1}}.\frac{\sqrt{\mathrm{31}}}{\mathrm{4}}\mathrm{du} \\ $$$$=\frac{\mathrm{16}}{\:\sqrt{\mathrm{31}}}\:\mathrm{arctan}\left(\mathrm{u}\right)\:+\mathrm{C}\:=\frac{\mathrm{16}}{\:\sqrt{\mathrm{31}}}\mathrm{arctan}\left(\frac{\mathrm{4x}+\mathrm{3}}{\:\sqrt{\mathrm{31}}}\right)\:+\mathrm{C} \\ $$
Commented by aurpeyz last updated on 25/Nov/20
$${thank}\:{you} \\ $$