find lim_(ξ→0) ∫_0 ^1 (dx/((√(1+ξx^2 ))−(√(1−ξx^2 ))))


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Question Number 39838 by math khazana by abdo last updated on 12/Jul/18

$f i n d {l i m}_{ξ \to 0} \int_{0}^{1} \frac{d x}{\sqrt{1 + ξ x^{2}} - \sqrt{1 - ξ x^{2}}}$
Commented by maxmathsup by imad last updated on 24/Jul/18
$let I(ξ) = ∫_0 ^1 (dx/((√(1+ξx^2 ))−(√(1−ξx^2 )))) I(ξ) = ∫_0 ^1 (((√(1+ξx^2 ))+(√(1−ξx^2 )))/(2ξx^2 )) dx = (1/(2ξ)) ∫_0 ^1 ((√(1+ξx^2 ))/x^2 )dx +(1/(2ξ)) ∫_0 ^1 ((√(1−ξx^2 ))/x^2 ) dx but (√(1+ξx^2 ))=(1+ξx^2 )^(1/2) =1+(1/2)((ξx^2 )/(1!)) (((1/2)((1/2)−1))/(2!)) (ξx^2 )^2 +o(ξ) =1+((ξx^2 )/2) −(1/8) ξ^2 x^4 +o(ξ) ⇒ ((√(1+ξx^2 ))/x^2 ) =(1/x^2 ) +(ξ/2) −(1/8) ξ^2 x^2 +o(ξ) also ((√(1−ξx^2 ))/x^2 ) = (1/x^2 ) −(ξ/2) +(1/8) ξ^2 x^2 +o(ξ) ⇒⇒(((√(1+ξx^2 ))+(√(1−ξx^2 )))/(2ξx^2 )) =(1/(2ξx^2 )){ (2/x^2 ) +o(ξ)} = (1/(ξ x^4 )) +o(ξ) ⇒ I(ξ) ∼ (1/ξ) ∫_0 ^1 (dx/x^4 ) =(1/ξ)[(1/(−4+1)) x^(−4+1) ]_0 ^1 =−(1/(3ξ))[ (1/x^3 )]_0 ^1 =∞$
$l e t I (ξ) = \int_{0}^{1} \frac{d x}{\sqrt{1 + ξ x^{2}} - \sqrt{1 - ξ x^{2}}}$ $I (ξ) = \int_{0}^{1} \frac{\sqrt{1 + ξ x^{2}} + \sqrt{1 - ξ x^{2}}}{2 ξ x^{2}} d x$ $= \frac{1}{2 ξ} \int_{0}^{1} \frac{\sqrt{1 + ξ x^{2}}}{x^{2}} d x + \frac{1}{2 ξ} \int_{0}^{1} \frac{\sqrt{1 - ξ x^{2}}}{x^{2}} d x b u t$ $\sqrt{1 + ξ x^{2}} = {(1 + ξ x^{2})}^{\frac{1}{2}} = 1 + \frac{1}{2} \frac{ξ x^{2}}{1!} \frac{\frac{1}{2} (\frac{1}{2} - 1)}{2!} {(ξ x^{2})}^{2} + o (ξ)$ $= 1 + \frac{ξ x^{2}}{2} - \frac{1}{8} ξ^{2} x^{4} + o (ξ) \Rightarrow \frac{\sqrt{1 + ξ x^{2}}}{x^{2}} = \frac{1}{x^{2}} + \frac{ξ}{2} - \frac{1}{8} ξ^{2} x^{2} + o (ξ) a l s o$ $\frac{\sqrt{1 - ξ x^{2}}}{x^{2}} = \frac{1}{x^{2}} - \frac{ξ}{2} + \frac{1}{8} ξ^{2} x^{2} + o (ξ) \Rightarrow\Rightarrow \frac{\sqrt{1 + ξ x^{2}} + \sqrt{1 - ξ x^{2}}}{2 ξ x^{2}}$ $= \frac{1}{2 ξ x^{2}} {\frac{2}{x^{2}} + o (ξ)} = \frac{1}{ξ x^{4}} + o (ξ) \Rightarrow I (ξ) \sim \frac{1}{ξ} \int_{0}^{1} \frac{d x}{x^{4}} = \frac{1}{ξ} {[\frac{1}{- 4 + 1} x^{- 4 + 1}]}_{0}^{1}$ $= - \frac{1}{3 ξ} {[\frac{1}{x^{3}}]}_{0}^{1} = \infty$
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