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Question Number 86328 by M±th+et£s last updated on 28/Mar/20

∫((x^2 −1)/((x^2 +1)(√(1+x^4 )))) dx

$$\int\frac{{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }}\:{dx} \\ $$

Answered by TANMAY PANACEA. last updated on 28/Mar/20

∫((1−(1/x^2 ))/(1+(1/x^2 )×x(√(x^2 +(1/x^2 )))))dx ∫((d(x+(1/x)))/((x+(1/x))(√((x+(1/x))^2 −2))))dx ∫(dk/(k(√(k^2 −2)))) k=(√2) secθ ∫(((√2) secθtanθdθ)/((√2) secθ.(√2) tanθ)) (1/(√2))∫dθ=(θ/(√2))+c (1/(√2))sec^(−1) ((k/(√2)))+c (1/(√2))sec^(−1) (((x+(1/x))/(√2)))+c

$$\int\frac{\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}{\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }×{x}\sqrt{{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}}{dx} \\ $$$$\int\frac{{d}\left({x}+\frac{\mathrm{1}}{{x}}\right)}{\left({x}+\frac{\mathrm{1}}{{x}}\right)\sqrt{\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} −\mathrm{2}}}{dx} \\ $$$$\int\frac{{dk}}{{k}\sqrt{{k}^{\mathrm{2}} −\mathrm{2}}} \\ $$$${k}=\sqrt{\mathrm{2}}\:{sec}\theta \\ $$$$\int\frac{\sqrt{\mathrm{2}}\:{sec}\theta{tan}\theta{d}\theta}{\sqrt{\mathrm{2}}\:{sec}\theta.\sqrt{\mathrm{2}}\:{tan}\theta} \\ $$$$\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\int{d}\theta=\frac{\theta}{\sqrt{\mathrm{2}}}+{c} \\ $$$$\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}{sec}^{−\mathrm{1}} \left(\frac{{k}}{\sqrt{\mathrm{2}}}\right)+{c} \\ $$$$\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}{sec}^{−\mathrm{1}} \left(\frac{{x}+\frac{\mathrm{1}}{{x}}}{\sqrt{\mathrm{2}}}\right)+{c} \\ $$$$ \\ $$$$ \\ $$

Commented by M±th+et£s last updated on 28/Mar/20

god bless you sir

$${god}\:{bless}\:{you}\:{sir} \\ $$

Commented by TANMAY PANACEA. last updated on 28/Mar/20

now covid 19 cegulfed the world...so god may bless to kill the virus...

$${now}\:{covid}\:\mathrm{19}\:{cegulfed}\:{the}\:{world}...{so}\:{god}\:{may}\:{bless} \\ $$$${to}\:{kill}\:{the}\:{virus}... \\ $$

Commented by niroj last updated on 28/Mar/20

Yes sir, Now the time is very creatical situation entire over world So we must prey our god even each religion to find early medicine from world medical team. Then it will be welfare for all public health and peacefull life. Please all ,dear sir follow the strickly dailly health protection for your safety and also follow the respectively lock down area when government are order in each country whatever we are now. Some days we are suffereing we will not die but if we destroy the rule may be chances to effected by virus. So that be safety and awareness all the people from COVID−19. thank you.

$$\:\:\mathrm{Yes}\:\mathrm{sir},\:\mathrm{Now}\:\mathrm{the}\:\mathrm{time}\:\mathrm{is}\:\mathrm{very}\:\mathrm{creatical}\:\mathrm{situation}\:\mathrm{entire}\:\mathrm{over}\:\mathrm{world} \\ $$$$\:\mathrm{So}\:\mathrm{we}\:\mathrm{must}\:\mathrm{prey}\:\mathrm{our}\:\mathrm{god}\:\mathrm{even}\:\mathrm{each}\:\mathrm{religion}\:\mathrm{to}\:\mathrm{find}\:\mathrm{early}\:\mathrm{medicine}\:\mathrm{from}\:\mathrm{world}\:\mathrm{medical}\:\mathrm{team}. \\ $$$$\:\mathrm{Then}\:\mathrm{it}\:\mathrm{will}\:\mathrm{be}\:\mathrm{welfare}\:\mathrm{for}\:\mathrm{all}\:\mathrm{public}\:\mathrm{health}\:\mathrm{and}\:\mathrm{peacefull}\:\mathrm{life}. \\ $$$$\:\mathrm{Please}\:\mathrm{all}\:,\mathrm{dear}\:\mathrm{sir}\:\:\mathrm{follow}\:\mathrm{the}\:\mathrm{strickly}\:\mathrm{dailly}\:\mathrm{health}\:\mathrm{protection}\:\mathrm{for}\:\mathrm{your}\:\mathrm{safety}\:\mathrm{and} \\ $$$$\:\mathrm{also}\:\mathrm{follow}\:\mathrm{the}\:\mathrm{respectively}\:\mathrm{lock}\:\mathrm{down}\:\mathrm{area}\:\mathrm{when}\:\mathrm{government}\:\mathrm{are}\:\mathrm{order}\:\mathrm{in}\:\mathrm{each}\:\mathrm{country}\:\mathrm{whatever}\:\mathrm{we}\:\mathrm{are}\:\mathrm{now}. \\ $$$$\:\:\mathrm{Some}\:\mathrm{days}\:\mathrm{we}\:\mathrm{are}\:\mathrm{suffereing}\:\mathrm{we}\:\mathrm{will}\:\mathrm{not}\:\mathrm{die}\:\mathrm{but}\:\mathrm{if}\:\mathrm{we}\:\mathrm{destroy}\:\:\mathrm{the}\:\mathrm{rule}\:\mathrm{may}\:\mathrm{be}\:\mathrm{chances}\:\mathrm{to}\:\mathrm{effected}\:\mathrm{by}\:\mathrm{virus}. \\ $$$$\:\:\mathrm{So}\:\mathrm{that}\:\mathrm{be}\:\mathrm{safety}\:\mathrm{and}\:\mathrm{awareness} \\ $$$$\mathrm{all}\:\mathrm{the}\:\mathrm{people}\:\:\mathrm{from}\:\boldsymbol{\mathrm{C}}\mathrm{O}\boldsymbol{\mathrm{VID}}−\mathrm{19}. \\ $$$$\:\:\mathrm{thank}\:\mathrm{you}. \\ $$$$ \\ $$

Commented by peter frank last updated on 29/Mar/20

thank you

$${thank}\:{you} \\ $$

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