∫_0 ^∞ e^t^2 e^(−st) dt please help


Question and Answers Forum

All Questions Topic List
Others Questions
Previous in All Question Next in All Question
Previous in Others Next in Others
Question Number 6775 by Tawakalitu. last updated on 24/Jul/16

$\int_{0}^{\infty} e^{t^{2}} e^{- s t} d t$ $p l e a s e h e l p$
	Answered by Yozzii last updated on 24/Jul/16
	$b=∫_0 ^∞ e^(t^2 −st) dt=∫_0 ^∞ e^((t−0.5s)^2 −0.25s^2 ) dt b=e^(−0.25s^2 ) ∫_0 ^∞ e^((t−0.5s)^2 ) dt. u=t−0.5s⇒du=dt ∴b=e^(−0.25s^2 ) (∫_(−0.5s) ^0 e^u^2 du+∫_0 ^∞ e^u^2 du) ∫_0 ^∞ e^u^2 du is divergent while ∫_(−0.5s) ^0 e^u^2 du is finite. ∴ For nonzero s∈R, b=e^(−0.25s^2 ) {∫_(−0.5s) ^0 e^u^2 du+∫_0 ^∞ e^u^2 du} does not exist. If s=0,b=∫_0 ^∞ e^t^2 dt which is divergent also.$
	$b = \int_{0}^{\infty} e^{t^{2} - s t} d t = \int_{0}^{\infty} e^{{(t - 0.5 s)}^{2} - 0.25 s^{2}} d t$ $b = e^{- 0.25 s^{2}} \int_{0}^{\infty} e^{{(t - 0.5 s)}^{2}} d t .$ $u = t - 0.5 s \Rightarrow d u = d t$ $∴ b = e^{- 0.25 s^{2}} (\int_{- 0.5 s}^{0} e^{u^{2}} d u + \int_{0}^{\infty} e^{u^{2}} d u)$ $\int_{0}^{\infty} e^{u^{2}} d u i s d i v e r g e n t w h i l e \int_{- 0.5 s}^{0} e^{u^{2}} d u$ $i s f i n i t e .$ $∴ F o r n o n z e r o s \in R, b = e^{- 0.25 s^{2}} {\int_{- 0.5 s}^{0} e^{u^{2}} d u + \int_{0}^{\infty} e^{u^{2}} d u}$ $d o e s n o t e x i s t .$ $I f s = 0, b = \int_{0}^{\infty} e^{t^{2}} d t w h i c h i s d i v e r g e n t a l s o .$
	Commented by Tawakalitu. last updated on 24/Jul/16

	$T h a n k s s o m u c h, T h a t m e a n s t h e q u e a s t i o n i s w r o n g ? ?$
	Commented by Yozzii last updated on 24/Jul/16

	$N o t n e c e s s a r i l y . T h e q u e s t i o n m a y h a v e$ $a s k e d t o s e e w h e t h e r t h e i m p r o p e r$ $i n t e g r a l i s c o n v e r g e n t o r d i v e r g e n t .$
	Commented by Tawakalitu. last updated on 24/Jul/16

	$T h a n k s$
	Commented by Tawakalitu. last updated on 25/Jul/16

	$T h e q u e s t i o n i s e v a l u a t e$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum