anonymous
  • anonymous
\int _{ -\infty }^{ \infty }{ \frac { { e }^{ -y } }{ { y }^{ 2 }+1 } } converges \quad or\quad diverge?
OCW Scholar - Single Variable Calculus
chestercat
  • chestercat
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anonymous
  • anonymous
\[\int\limits _{ -\infty }^{ \infty }{ \frac { { e }^{ -y } }{ { y }^{ 2 }+1 } } converges \quad or\quad diverege? \]
anonymous
  • anonymous
Diverges because from minus infinity to zero the expression e^-y is much much larger than y^2+1. Actually because you take the integral from minus infinity to infinity this holds:\[\int\limits_{-\infty}^{\infty}\frac{ e^-y }{ y^2+1 }dy = \int\limits_{-\infty}^{\infty}\frac{ e^y }{ y^2+1 }dy \]
anonymous
  • anonymous
diverges. take simple example of \[\int\limits_{-infinity}^{infinity}(e^x)dx is \not convergent. and whether you divide \it by X^2 +1 .\] that is a positive no....wont effect as we are talkin about infinity/infinty

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