Calculus-dx-x-2-e-a-x-2-a-gt-0-

Question Number 143735 by mnjuly1970 last updated on 17/Jun/21

.....Calculus..... Ω:= ∫_(−∞) ^( ∞) (dx/(x^( 2) e^(a/x^2 ) )) =? (a > 0 )

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…..{Calculus}….. \\ $$$$\:\:\:\:\:\:\:\:\Omega:=\:\int_{−\infty} ^{\:\infty} \frac{{dx}}{{x}^{\:\mathrm{2}} \:{e}^{\frac{{a}}{{x}^{\mathrm{2}} }} }\:=?\:\:\left({a}\:>\:\mathrm{0}\:\right) \\ $$

Answered by Dwaipayan Shikari last updated on 17/Jun/21

(1/(x ))=u =2∫_0 ^∞ e^(−au^2 ) du =(√(π/a))

$$\frac{\mathrm{1}}{{x}\:}={u} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\infty} {e}^{−{au}^{\mathrm{2}} } {du}\:\:\:\:\:\: \\ $$$$=\sqrt{\frac{\pi}{{a}}} \\ $$

Commented by Dwaipayan Shikari last updated on 17/Jun/21