Question Number 122940 by CanovasCamiseros last updated on 21/Nov/20
Commented by CanovasCamiseros last updated on 21/Nov/20
$$\boldsymbol{{how}}\:\boldsymbol{{can}}\:\boldsymbol{{i}}\:\boldsymbol{{find}}\:\boldsymbol{{it}}? \\ $$
Commented by mr W last updated on 21/Nov/20
$${go}\:{to}\:{Q}\mathrm{73012} \\ $$
Answered by TANMAY PANACEA last updated on 21/Nov/20
$${t}^{\mathrm{2}} ={tanx}\rightarrow\mathrm{2}{t}\frac{{dt}}{{dx}}={sec}^{\mathrm{2}} {x} \\ $$$${dx}=\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{4}} }{dt} \\ $$$$\int\frac{{t}×\mathrm{2}{tdt}}{\mathrm{1}+{t}^{\mathrm{4}} } \\ $$$$\int\frac{\mathrm{2}{dt}}{{t}^{\mathrm{2}} +\frac{\mathrm{1}}{{t}^{\mathrm{2}} }} \\ $$$$\int\frac{\mathrm{1}−\frac{\mathrm{1}}{{t}^{\mathrm{2}} }+\mathrm{1}+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }}{{t}^{\mathrm{2}} +\frac{\mathrm{1}}{{t}^{\mathrm{2}} }}{dt} \\ $$$$\int\frac{{d}\left({t}+\frac{\mathrm{1}}{{t}}\right)}{\left({t}+\frac{\mathrm{1}}{{t}}\right)^{\mathrm{2}} −\mathrm{2}}+\int\frac{{d}\left({t}−\frac{\mathrm{1}}{{t}}\right)}{\left({t}−\frac{\mathrm{1}}{{t}}\right)^{\mathrm{2}} +\mathrm{2}} \\ $$$${use}\:{formula}\int\frac{{dx}}{{x}^{\mathrm{2}} −{a}^{\mathrm{2}} }\:\&\frac{\int{dx}}{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} } \\ $$$$\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}{ln}\left(\frac{\left.{t}+\frac{\mathrm{1}}{{t}}\right)−\sqrt{\mathrm{2}}}{\left({t}+\frac{\mathrm{1}}{{t}}+\sqrt{\mathrm{2}}\right.}\right)+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}{tan}^{−\mathrm{1}} \left(\frac{{t}−\frac{\mathrm{1}}{{t}}}{\:\sqrt{\mathrm{2}}}\right) \\ $$$${puy}\:\:\:{t}=\sqrt{{tanx}}\: \\ $$$$ \\ $$$$ \\ $$