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
$$F(x)=\int_a^xf(t)dt,$$
then F is differentiable on (a,b) with F'(x)=f(x) for all x in (a,b), i.e.
$$\frac{d}{dx}\int_a^x f(t)dt=f(x).$$Question: Is it correct also to write
$$\frac{d}{dx}\int f(t)dt=f(x)?$$

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
$$\frac{d}{dx}\int f(t)dt=f(x)?$$

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.