Fundemental Theorem of Calculus

[Niles]

Homework Statement

Hi guys

From the Fundemental Theorem of Calculus we know that if we have a continuous function f : [a,b] -> R and F is the function on (a,b) defined by
<br /> F(x)=\int_a^xf(t)dt,<br />
then F is differentiable on (a,b) with F'(x)=f(x) for all x in (a,b), i.e.
<br /> \frac{d}{dx}\int_a^x f(t)dt=f(x).<br />Question: Is it correct also to write
<br /> \frac{d}{dx}\int f(t)dt=f(x)?<br />

If not, then is there a way of expressing \frac{d}{dx}\int_a^x f(t)dt=f(x) without limits on the integral?

 

[ystael]

Question: Is it correct also to write
<br /> \frac{d}{dx}\int f(t)dt=f(x)?<br />

When you write that, everyone knows what you mean -- but it's not correct. The reason is that
\int f(t)\,dt
doesn't name a function of x, even though some dependence on x is clearly intended. You could think of it as naming a one-parameter family of functions
F_a(x) = \int_a^x f(t)\,dt
where the parameter a is the lower limit of the integral. The parameter shows up in another guise as the customary "constant of integration" C, which is connected with the fact that F_b - F_a is the constant function equal to
\int_a^b f(t)\,dt.

The functions F_a all differ by a constant, so they have the same derivative, which is f by the fundamental theorem. But the one-parameter family of functions \{F_a : a \in \mathbb{R}\} isn't something you can use the same notation to differentiate.

 

[HallsofIvy]

Occaisionally, you will see the notation
\int^x f(t)dt
(note the lack of a lower limit) to indicate the general anti-derivative.