prove-lim-n-0-1-n-1-n-2-x-2-e-x-2-dx-pi-2-

Question Number 213169 by MrGaster last updated on 31/Oct/24

p r o v e

\lim_{n \to \infty} \int_{0}^{1} \frac{n}{1 + n^{2} x^{2}} e^{x^{2}} d x = \frac{π}{2} .

Answered by Berbere last updated on 31/Oct/24

\int_{>}^{1} \frac{n}{1 + n^{2} x^{2}} e^{x^{2}} d x = [\tan^{- 1} (n x) e^{x^{2}}] - \int_{0}^{1} 2 {x e}^{x^{2}} \tan^{- 1} (n x) d x

= \frac{π}{2} e - \int_{0}^{1} 2 {x e}^{x^{2}} \tan^{- 1} (n x) d x = \frac{π e}{2} - \int_{0}^{1} 2 {x e}^{x^{2}} [\frac{π}{2} - \tan^{- 1} (\frac{1}{n x})] d x

= \frac{π e}{2} - \frac{π}{2} (e - 1) + 2 \int_{0}^{1} {x e}^{x^{2}} \tan^{- 1} (\frac{1}{n x})

= \frac{π}{2} + B

\forall x ⩾ 0 \tan^{- 1} (x) < x

\Rightarrow \int_{0}^{1} {x e}^{x^{2}} \tan^{- 1} (\frac{1}{n x}) d x ⩽ \int_{0}^{1} \frac{e^{x^{2}}}{n} d x ⩽ \frac{e}{n}

\Rightarrow \lim_{n \to \infty} B = 0 \Rightarrow

\lim_{n \to \infty} \int_{0}^{1} \frac{n}{1 + n^{2} x^{2}} e^{x^{2}} d x = \frac{π}{2}

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