let-F-x-0-e-x-2-t-t-1-t-2-dt-calculate-lim-x-F-x-

Question Number 36189 by prof Abdo imad last updated on 30/May/18

l e t F (x) = \int_{0}^{\infty} \frac{e^{- x^{2} t} \sqrt{t}}{1 + t^{2}} d t

c a l c u l a t e {l i m}_{x \to + \infty} F (x) .

Commented by math khazana by abdo last updated on 18/Aug/18

c h a n g e m e n t \sqrt{t} = u g i v e

F (x) = \int_{0}^{\infty} \frac{e^{- x^{2} u^{2}} u}{1 + u^{4}} (2 u) d u

= 2 \int_{0}^{\infty} \frac{u^{2} e^{- x^{2} u^{2}}}{1 + u^{4}} d u

=_{x u = α} 2 \int_{0}^{\infty} \frac{α^{2}}{x^{2}} \frac{e^{- α^{2}}}{1 + \frac{α^{4}}{x^{4}}} \frac{1}{x} d α

= 2 \int_{0}^{\infty} \frac{α^{2} e^{- α^{2}}}{x^{3} + \frac{α^{4}}{x}} d α = 2 x \int_{0}^{\infty} \frac{α^{2} e^{- α^{2}}}{x^{4} + α^{4}} d α

\Rightarrow F (x) ⩽ \frac{2}{x^{3}} \int_{0}^{\infty} α^{2} e^{- α^{2}} d α b u t \int_{0}^{\infty} α^{2} e^{- α^{2}} d α

c o n v e r g e s \Rightarrow {l i m}_{x \to + \infty} F (x) = 0 .