lim_( n→ ∞) ∫_0 ^( 1) (( n . e^( 1− x^( 2) ) )/( 1 + n^( 2) x^( 2) )) dx =? −−−−−−


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Question Number 165273 by mnjuly1970 last updated on 28/Jan/22

${l i m}_{n \to \infty} \int_{0}^{1} \frac{n . e^{1 - x^{2}}}{1 + n^{2} x^{2}} d x = ?$ $- - - - - -$
	Answered by mindispower last updated on 28/Jan/22

	$\int \frac{n}{1 + n^{2} x^{2}} d x = a r c t a n (n x)$ $\lim_{n \to \infty} {[a r c t a n (n x) e^{1 - x^{2}}]}_{0}^{1} + 2 \int_{0}^{1} {x e}^{1 - x^{2}} a r c t a n (n x) d x$ $= \underset{n \to \infty}{\lim 2} \int_{0}^{1} a r c t a n (n x) . {x e}^{1 - x^{2}} d x$ $2 \int_{0}^{1} x \tan^{- 1} (n x) e^{1 - x^{2}} d x ⩽ 2 \int_{0}^{1} \frac{π}{2} {x e}^{1 - x^{2}} = \frac{π}{2}$ $w e U s e ‘ ‘ T h e o r e m d e C o n e r g e n c e {D o m i n e e}^{″}$ $i t s T h e N a m e i n f r a n c e i n E n g l i s h I d o n t K n o w$ $H o w W e C a l l i T$ $\underset{n \to \infty}{\lim 2} \int_{0}^{1} x \tan^{- 1} (n x) e^{1 - x^{2}} d x = \int_{0}^{1} {\underset{n \to \infty}{\lim 2 t a n}}^{- 1} (n x) . {x e}^{1 - x^{2}} d x$ $= \int_{0}^{1} \frac{π}{2} (2 {x e}^{1 - x^{2}}) d x . \frac{π}{2} {[- e^{1 - x^{2}}]}_{0}^{1} = \frac{π}{2} e$
	Commented by mnjuly1970 last updated on 28/Jan/22

	$b r a v o, s i r p o w e r . . . . v e r y n i c e$ $s o l u t i o n$ $D o m i n a t i n g c o n v e r g e n c e t h e o r e m . . .$
	Commented by mindispower last updated on 02/Feb/22

	$T h a n x S i r$ $H a v e a N i c e D a y$
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