1. ## Simpl-integral

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

2. 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 $b = 0$

$= \sqrt{2}$