# An extension of the Fundamental Theorem of Calculus

Castilla
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.

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.

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

Castilla