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Question-43417

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Question Number 43417 by Raj Singh last updated on 10/Sep/18
Commented by alex041103 last updated on 10/Sep/18
is that ((x^n +a^n ))^(1/x)
$${is}\:{that}\:\sqrt[{{x}}]{{x}^{{n}} +{a}^{{n}} } \\ $$
Commented by maxmathsup by imad last updated on 10/Sep/18
let I_n = ∫ (dx/(x(√(x^n +a^n )))) ⇒ I_n = ∫ (dx/(x a^(n/2) (√(((x/a))^n +1)))) changement ((x/a))^n =tan^2 θ give (x/a)=(tan^2 θ)^(1/n) ⇒ dx =(a/n) 2tan(θ)(1+tan^2 θ)(tan^2 θ)^((1/n)−1) dθ I_n =((2a)/n) ∫ ((tanθ (1+tan^2 θ)(tan^2 θ)^((1/n)−1) )/(a (tan^2 θ)^(1/n) (√(1+tan^2 θ)))) dθ =(2/n) ∫ tan^(−1) θ (√(1+tan^2 θ))dθ =(2/n) ∫ (1/(cos^2 θ ((sinθ)/(cosθ)))) dθ =(2/n) ∫ ((2dθ)/(sin(2θ))) =_(2θ=u) (2/n) ∫ (du/(sinu)) =_(tan((u/2))=α) (2/n) ∫ (1/((2α)/(1+α^2 ))) ((2dα)/(1+α^2 )) =(2/(n )) ∫ (dα/α) =(2/n)ln∣α∣ +c =(2/n)ln∣tan((u/2))∣ =(2/n)ln∣tanθ∣ =(2/n)ln∣tan( arctan((x/a))^(n/2) )∣ +c .
$${let}\:{I}_{{n}} =\:\int\:\:\:\:\frac{{dx}}{{x}\sqrt{{x}^{{n}} \:+{a}^{{n}} }}\:\Rightarrow\:{I}_{{n}} =\:\int\:\:\:\:\:\frac{{dx}}{{x}\:{a}^{\frac{{n}}{\mathrm{2}}} \sqrt{\left(\frac{{x}}{{a}}\right)^{{n}} +\mathrm{1}}}\:\:{changement}\:\:\left(\frac{{x}}{{a}}\right)^{{n}} ={tan}^{\mathrm{2}} \theta \\ $$$${give}\:\frac{{x}}{{a}}=\left({tan}^{\mathrm{2}} \theta\right)^{\frac{\mathrm{1}}{{n}}} \:\Rightarrow\:{dx}\:=\frac{{a}}{{n}}\:\mathrm{2}{tan}\left(\theta\right)\left(\mathrm{1}+{tan}^{\mathrm{2}} \theta\right)\left({tan}^{\mathrm{2}} \theta\right)^{\frac{\mathrm{1}}{{n}}−\mathrm{1}} \:{d}\theta \\ $$$${I}_{{n}} =\frac{\mathrm{2}{a}}{{n}}\:\int\:\:\:\:\:\:\:\frac{{tan}\theta\:\left(\mathrm{1}+{tan}^{\mathrm{2}} \theta\right)\left({tan}^{\mathrm{2}} \theta\right)^{\frac{\mathrm{1}}{{n}}−\mathrm{1}} }{{a}\:\left({tan}^{\mathrm{2}} \theta\right)^{\frac{\mathrm{1}}{{n}}} \:\sqrt{\mathrm{1}+{tan}^{\mathrm{2}} \theta}}\:{d}\theta\: \\ $$$$=\frac{\mathrm{2}}{{n}}\:\int\:\:{tan}^{−\mathrm{1}} \theta\:\sqrt{\mathrm{1}+{tan}^{\mathrm{2}} \theta}{d}\theta\:=\frac{\mathrm{2}}{{n}}\:\int\:\:\:\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \theta\:\frac{{sin}\theta}{{cos}\theta}}\:{d}\theta \\ $$$$=\frac{\mathrm{2}}{{n}}\:\int\:\:\:\frac{\mathrm{2}{d}\theta}{{sin}\left(\mathrm{2}\theta\right)}\:=_{\mathrm{2}\theta={u}} \:\:\frac{\mathrm{2}}{{n}}\:\int\:\:\frac{{du}}{{sinu}}\:=_{{tan}\left(\frac{{u}}{\mathrm{2}}\right)=\alpha} \:\frac{\mathrm{2}}{{n}}\:\int\:\:\:\:\frac{\mathrm{1}}{\frac{\mathrm{2}\alpha}{\mathrm{1}+\alpha^{\mathrm{2}} }}\:\frac{\mathrm{2}{d}\alpha}{\mathrm{1}+\alpha^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{2}}{{n}\:}\:\int\:\:\frac{{d}\alpha}{\alpha}\:=\frac{\mathrm{2}}{{n}}{ln}\mid\alpha\mid\:+{c}\:=\frac{\mathrm{2}}{{n}}{ln}\mid{tan}\left(\frac{{u}}{\mathrm{2}}\right)\mid\:=\frac{\mathrm{2}}{{n}}{ln}\mid{tan}\theta\mid \\ $$$$=\frac{\mathrm{2}}{{n}}{ln}\mid{tan}\left(\:{arctan}\left(\frac{{x}}{{a}}\right)^{\frac{{n}}{\mathrm{2}}} \right)\mid\:\:+{c}\:. \\ $$$$ \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 10/Sep/18
∫(dx/(x(√(x^n +a^n )))) ∫((x^(n−1) dx)/(x^n (√(x^n +a^n )))) t^2 =x^n +a^n 2tdt=nx^(n−1) dx ∫((2tdt)/(n×(t^2 −a^n )×t)) (2/n)∫(dt/(t^2 −(a^(n/2) )^2 )) (2/n)×(1/(2a^(n/2) ))∫(((t+a^(n/2) )−(t−a^(n/2) ))/((t+a^(n/2) )(t−a^(n/2) )))dt =(1/(na^(n/2) ))[∫(dt/(t−a^(n/2) ))−∫(dt/(t+a^(n/2) ))] =(1/(na^(n/2) ))ln∣((t−a^(n/2) )/(t+a^(n/2) ))∣+c (1/(na^(n/2) ))ln∣(((√(x^n +a^n )) −a^(n/2) )/( (√(x^n +a^n )) +a^(n/2) ))∣+c
$$\int\frac{{dx}}{{x}\sqrt{{x}^{{n}} +{a}^{{n}} }} \\ $$$$\int\frac{{x}^{{n}−\mathrm{1}} {dx}}{{x}^{{n}} \sqrt{{x}^{{n}} +{a}^{{n}} }} \\ $$$${t}^{\mathrm{2}} ={x}^{{n}} +{a}^{{n}} \\ $$$$\mathrm{2}{tdt}={nx}^{{n}−\mathrm{1}} {dx} \\ $$$$\int\frac{\mathrm{2}{tdt}}{{n}×\left({t}^{\mathrm{2}} −{a}^{{n}} \right)×{t}} \\ $$$$\frac{\mathrm{2}}{{n}}\int\frac{{dt}}{{t}^{\mathrm{2}} −\left({a}^{\frac{{n}}{\mathrm{2}}} \right)^{\mathrm{2}} } \\ $$$$\frac{\mathrm{2}}{{n}}×\frac{\mathrm{1}}{\mathrm{2}{a}^{\frac{{n}}{\mathrm{2}}} }\int\frac{\left({t}+{a}^{\frac{{n}}{\mathrm{2}}} \right)−\left({t}−{a}^{\frac{{n}}{\mathrm{2}}} \right)}{\left({t}+{a}^{\frac{{n}}{\mathrm{2}}} \right)\left({t}−{a}^{\frac{{n}}{\mathrm{2}}} \right)}{dt} \\ $$$$=\frac{\mathrm{1}}{{na}^{\frac{{n}}{\mathrm{2}}} }\left[\int\frac{{dt}}{{t}−{a}^{\frac{{n}}{\mathrm{2}}} }−\int\frac{{dt}}{{t}+{a}^{\frac{{n}}{\mathrm{2}}} }\right] \\ $$$$=\frac{\mathrm{1}}{{na}^{\frac{{n}}{\mathrm{2}}} }{ln}\mid\frac{{t}−{a}^{\frac{{n}}{\mathrm{2}}} }{{t}+{a}^{\frac{{n}}{\mathrm{2}}} }\mid+{c} \\ $$$$\frac{\mathrm{1}}{{na}^{\frac{{n}}{\mathrm{2}}} }{ln}\mid\frac{\sqrt{{x}^{{n}} +{a}^{{n}} }\:−{a}^{\frac{{n}}{\mathrm{2}}} }{\:\sqrt{{x}^{{n}} +{a}^{{n}} }\:+{a}^{\frac{{n}}{\mathrm{2}}} }\mid+{c} \\ $$

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