Question Number 86258 by lémùst last updated on 27/Mar/20
$${calculate}\:{I}=\int_{\mathrm{1}} ^{+\infty} \frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{4}} −{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$
Commented by mathmax by abdo last updated on 27/Mar/20
$${I}\:=\int_{\mathrm{1}} ^{+\infty} \:\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{4}} −{x}^{\mathrm{2}} \:+\mathrm{1}}{dx}\:\:\:\left({complex}\:{method}\right) \\ $$$${x}^{\mathrm{4}} −{x}^{\mathrm{2}} \:+\mathrm{1}\:=\mathrm{0}\:\rightarrow{t}^{\mathrm{2}} −{t}+\mathrm{1}\:=\mathrm{0}\:\:\left({t}={x}^{\mathrm{2}} \right) \\ $$$$\Delta=\mathrm{1}−\mathrm{4}=−\mathrm{3}\:\Rightarrow{t}_{\mathrm{1}} =\frac{\mathrm{1}+{i}\sqrt{\mathrm{3}}}{\mathrm{2}}={e}^{\frac{{i}\pi}{\mathrm{6}}} \:\:{and}\:{t}_{\mathrm{2}} =\frac{\mathrm{1}−{i}\sqrt{\mathrm{3}}}{\mathrm{2}}={e}^{−\frac{{i}\pi}{\mathrm{6}}} \\ $$$$\Rightarrow\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{4}} −{x}^{\mathrm{2}} \:+\mathrm{1}}\:=\frac{{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}^{\mathrm{2}} −{e}^{\frac{{i}\pi}{\mathrm{6}}} \right)\left({x}^{\mathrm{2}} \:−{e}^{−\frac{{i}\pi}{\mathrm{6}}} \right)}\:=\frac{{x}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{isin}\left(\frac{\pi}{\mathrm{6}}\right)}\left\{\frac{\mathrm{1}}{{x}^{\mathrm{2}} −{e}^{\frac{{i}\pi}{\mathrm{6}}} }−\frac{\mathrm{1}}{{x}^{\mathrm{2}} \:−{e}^{−\frac{{i}\pi}{\mathrm{6}}} }\right\} \\ $$$$=−{i}\left(\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{2}} −{e}^{\frac{{i}\pi}{\mathrm{6}}} }\:−\frac{{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{2}} \:−{e}^{−\frac{{i}\pi}{\mathrm{6}}} }\right)\:=−{i}\left(\frac{{x}^{\mathrm{2}} −{e}^{\frac{{i}\pi}{\mathrm{6}}} \:+{e}^{\frac{{i}\pi}{\mathrm{6}}} −\mathrm{1}}{{x}^{\mathrm{2}} −{e}^{\frac{{i}\pi}{\mathrm{6}}} }−\frac{{x}^{\mathrm{2}} −{e}^{−\frac{{i}\pi}{\mathrm{6}}} +{e}^{−\frac{{i}\pi}{\mathrm{6}}} −\mathrm{1}}{{x}^{\mathrm{2}} −{e}^{−\frac{{i}\pi}{\mathrm{6}}} }\right) \\ $$$$=−{i}\left\{\frac{{e}^{\frac{{i}\pi}{\mathrm{6}}} −\mathrm{1}}{{x}^{\mathrm{2}} −{e}^{\frac{{i}\pi}{\mathrm{6}}} }−\frac{{e}^{−\frac{{i}\pi}{\mathrm{6}}} −\mathrm{1}}{{x}^{\mathrm{2}} −{e}^{−\frac{{i}\pi}{\mathrm{6}}} }\right\}\:\Rightarrow{I}={i}\left({e}^{−\frac{{i}\pi}{\mathrm{6}}} −\mathrm{1}\right)\int_{\mathrm{1}} ^{+\infty} \:\frac{{dx}}{{x}^{\mathrm{2}} −{e}^{−\frac{{i}\pi}{\mathrm{6}}} } \\ $$$$−{i}\left({e}^{\frac{{i}\pi}{\mathrm{6}}} −\mathrm{1}\right)\int_{\mathrm{1}} ^{+\infty} \frac{{dx}}{{x}^{\mathrm{2}} −{e}^{\frac{{i}\pi}{\mathrm{6}}} } \\ $$$$\int_{\mathrm{1}} ^{+\infty} \:\frac{{dx}}{{x}^{\mathrm{2}} −{e}^{\frac{{i}\pi}{\mathrm{6}}} }\:=\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{dx}}{\left({x}−{e}^{\frac{{i}\pi}{\mathrm{12}}} \right)\left({x}+{e}^{\frac{{i}\pi}{\mathrm{12}}} \right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{e}^{−\frac{{i}\pi}{\mathrm{12}}} \int_{\mathrm{1}} ^{+\infty} \:\left(\frac{\mathrm{1}}{{x}−{e}^{\frac{{i}\pi}{\mathrm{12}}} }−\frac{\mathrm{1}}{{x}+{e}^{\frac{{i}\pi}{\mathrm{12}}} }\right){dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{e}^{−\frac{{i}\pi}{\mathrm{12}}} \left[{ln}\left(\frac{{x}−{e}^{\frac{{i}\pi}{\mathrm{12}}} }{{x}+{e}^{\frac{{i}\pi}{\mathrm{12}}} }\right)\right]_{\mathrm{1}} ^{+\infty} \:=\frac{\mathrm{1}}{\mathrm{2}}{e}^{−\frac{{i}\pi}{\mathrm{12}}} \left\{−{ln}\left(\frac{\mathrm{1}−{e}^{\frac{{i}\pi}{\mathrm{12}}} }{\mathrm{1}+{e}^{\frac{{i}\pi}{\mathrm{12}}} }\right)\right\} \\ $$$$…{becontinued}… \\ $$
Commented by lémùst last updated on 28/Mar/20
$${sir}\:{how}\:{do}\:{you}\:{calculate}\:{the}\:{logarith}\:{of}\:{a} \\ $$$${complex}\:? \\ $$
Commented by mathmax by abdo last updated on 31/Mar/20
$${ln}\left({x}+{iy}\right)\:={ln}\left(\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{e}^{{i}\:{arctan}\left(\frac{{y}}{{x}}\right)} \right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{ln}\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right)+{iarctan}\left(\frac{{y}}{{x}}\right)\:\:\left({principle}\:{determination}\:{of}\:{ln}\right) \\ $$
Answered by TANMAY PANACEA. last updated on 28/Mar/20
$$\int\frac{\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}{{x}^{\mathrm{2}} −\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}{dx} \\ $$$$\int\frac{{d}\left({x}+\frac{\mathrm{1}}{{x}}\right)}{\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} −\mathrm{3}}=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}{ln}\left(\frac{{x}+\frac{\mathrm{1}}{{x}}−\sqrt{\mathrm{3}}}{{x}+\frac{\mathrm{1}}{{x}}+\sqrt{\mathrm{3}}}\right) \\ $$$${useing}\:{formula}\:\int\frac{{dp}}{{p}^{\mathrm{2}} −{a}^{\mathrm{2}} } \\ $$$$ \\ $$
Commented by lémùst last updated on 28/Mar/20
$${thanks}\:{sir}\: \\ $$
Commented by TANMAY PANACEA. last updated on 28/Mar/20
$${most}\:{welcome}… \\ $$