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Question Number 42791 by maxmathsup by imad last updated on 02/Sep/18

find  ∫    (dx/((x^(2 ) +1)(√(1+x^2 ))))

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

Answered by MJS last updated on 02/Sep/18

I know this because of the hyperbola formuls  y=(√(x^2 +1))  y′=(x/(√(x^2 +1)))  y′′=(1/((x^2 +1)^(3/2) ))  ⇒ ∫(dx/((x^2 +1)(√(x^2 +1))))=∫(dx/((x^2 +1)^(3/2) ))=(x/(√(x^2 +1)))+C

$$\mathrm{I}\:\mathrm{know}\:\mathrm{this}\:\mathrm{because}\:\mathrm{of}\:\mathrm{the}\:\mathrm{hyperbola}\:\mathrm{formuls} \\ $$$${y}=\sqrt{{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$${y}'=\frac{{x}}{\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}} \\ $$$${y}''=\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\Rightarrow\:\int\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}=\int\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }=\frac{{x}}{\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}+{C} \\ $$

Commented by math khazana by abdo last updated on 03/Sep/18

thank you sir.

$${thank}\:{you}\:{sir}. \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Sep/18

x=tanα   dx=sec^2 α dα  ∫((sec^2 αdα)/((sec^2 α)^(3/2) ))  ∫cosαdα  sinα+c  =(x/(√(1+x^2 )))+c

$${x}={tan}\alpha\:\:\:{dx}={sec}^{\mathrm{2}} \alpha\:{d}\alpha \\ $$$$\int\frac{{sec}^{\mathrm{2}} \alpha{d}\alpha}{\left({sec}^{\mathrm{2}} \alpha\right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\int{cos}\alpha{d}\alpha \\ $$$${sin}\alpha+{c} \\ $$$$=\frac{{x}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}+{c} \\ $$

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