Simpl-integral

**dhiab** · May 28th 2010, 09:35 AM

If :
$\120dpi \int_{-\infty }^{\infty }e^{bt}f\left ( t \right )dt=sin^{-1}\left ( b-\frac{1}{\sqrt{2}} \right )$
Calculate :
$\120dpi \int_{-\infty }^{\infty }t\cdot f\left ( t \right )dt$

**Random Variable** · May 28th 2010, 11:22 AM

let $I(b) = \int^{\infty}_{-\infty} e^{bt} f(t) \ dt$

then $I'(b) = \int^{\infty}_{-\infty} t e^{bt} f(t) \ dt$

and $I'(0) = \int^{\infty}_{-\infty} t f(t) \ dt = \frac{d}{db} \arcsin \Big( b- \frac{1}{\sqrt{2}} \Big)$ evaluated at $b = 0$

$= \sqrt{2}$

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