# An extension of the Fundamental Theorem of Calculus

## Main Question or Discussion Point

In a book of Introduction to Probability I found this statement:

" Let be $$F(x) = \int_{-\infty}^{x} f(t)dt.$$ Then, by the Fundamental Theorem of Calculus, $$F'(x) = f(x).$$"

With the minus infinity on the lower limit, it is this a valid aplication of the FTC???

Thanks.

DrGreg
Gold Member
Castilla said:
In a book of Introduction to Probability I found this statement:

" Let be $$F(x) = \int_{-\infty}^{x} f(t)dt.$$ Then, by the Fundamental Theorem of Calculus, $$F'(x) = f(x).$$"

With the minus infinity on the lower limit, it is this a valid aplication of the FTC???

Thanks.

$$F(x) = \int_{-\infty}^{a} f(t)dt + \int_{a}^{x} f(t)dt$$

for any constant a < x. So the answer is yes.

Oh god, it was so easy...thanks, Greg.

Castilla