Question Number 166610 by cortano1 last updated on 23/Feb/22 | ||
$$\:\:\:\:\int\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{3}}{\mathrm{x}^{\mathrm{4}} +\mathrm{5x}^{\mathrm{2}} +\mathrm{9}}\:\mathrm{dx}\:? \\ $$ | ||
Commented by MJS_new last updated on 23/Feb/22 | ||
$$\mathrm{in}\:\mathrm{this}\:\mathrm{case}: \\ $$$$\frac{{d}}{{dx}}\left[\frac{\mathrm{1}}{{a}}\mathrm{arctan}\:\frac{{ax}}{{b}−{x}^{\mathrm{2}} }\right]=\frac{{x}^{\mathrm{2}} +{b}}{{x}^{\mathrm{4}} +\left({a}^{\mathrm{2}} −\mathrm{2}{b}\right){x}^{\mathrm{2}} +{b}^{\mathrm{2}} } \\ $$$$\mathrm{with}\:{a}=\sqrt{\mathrm{11}}\:\wedge\:{b}=\mathrm{3} \\ $$$$\Rightarrow\:\int\frac{{x}^{\mathrm{2}} +\mathrm{3}}{{x}^{\mathrm{4}} +\mathrm{5}{x}^{\mathrm{2}} +\mathrm{9}}{dx}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{11}}}\mathrm{arctan}\:\frac{\sqrt{\mathrm{11}}{x}}{\mathrm{3}−{x}^{\mathrm{2}} }\:+{C} \\ $$ | ||
Answered by MJS_new last updated on 23/Feb/22 | ||
$$\int\frac{{x}^{\mathrm{2}} +\mathrm{3}}{{x}^{\mathrm{4}} +\mathrm{5}{x}^{\mathrm{2}} +\mathrm{9}}{dx}=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{dx}}{{x}^{\mathrm{2}} −{x}+\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{dx}}{{x}^{\mathrm{2}} +{x}+\mathrm{3}}= \\ $$$$\:\:\:\:\:\left[\mathrm{use}\:\mathrm{commom}\:\mathrm{formula}\right] \\ $$$$=\frac{\sqrt{\mathrm{11}}}{\mathrm{11}}\left(\mathrm{arctan}\:\frac{\mathrm{2}{x}−\mathrm{1}}{\:\sqrt{\mathrm{11}}}\:+\mathrm{arctan}\:\frac{\mathrm{2}{x}+\mathrm{1}}{\:\sqrt{\mathrm{11}}}\right)+{C} \\ $$ | ||
Commented by cortano1 last updated on 25/Feb/22 | ||
$$\mathrm{nice} \\ $$ | ||