Simpl-integral

May 2009
596
31
ALGERIA
If :

Calculate :
 
May 2009
959
362
let \(\displaystyle I(b) = \int^{\infty}_{-\infty} e^{bt} f(t) \ dt \)

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

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

\(\displaystyle = \sqrt{2} \)
 
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