Limits of Integration Variable

[Jhenrique]

Knowing that the limits of integration of a any function, for example:
$$\int_{-\infty}^{+\infty}\delta (x)dx=1$$
I know that's correct call your primitive through the limit superior as a variable, so
$$H(x)=\int_{-\infty}^{x}\delta (x)dx$$
But, and if I want to describe your primitive through the limit inferior as a variable? Will be so:
$$H(x)=\int_{-x}^{+\infty}\delta (x)dx$$
or:
$$H(x)=\int_{+x}^{+\infty}\delta (x)dx$$
or other?

 

[Mark44]

Knowing that the limits of integration of a any function, for example:
$$\int_{-\infty}^{+\infty}\delta (x)dx=1$$

I am having a hard time understanding what you're saying and what you're asking. I can't tell if you're asking about some generic function whose integral is 1, or if your question is about the Dirac delta function (see http://en.wikipedia.org/wiki/Dirac_delta_function).

I know that's correct call your primitive through the limit superior as a variable, so
$$H(x)=\int_{-\infty}^{x}\delta (x)dx$$

Here H is a function of x. Clearly the value of H(x) is somewhere between 0 and 1.

But, and if I want to describe your primitive through the limit inferior as a variable? Will be so:
$$H(x)=\int_{-x}^{+\infty}\delta (x)dx$$
or:
$$H(x)=\int_{+x}^{+\infty}\delta (x)dx$$
or other?

 

[Jhenrique]

I saw that when you have a definite integral of a function f(x), you can to express the primitive, F(x), placing the variable x in limit superior of integral:
$$F(x)=\int_{x_{0}}^{x}f(x)dx$$
It's known... So I ask if F(x) can be equal to this too:
$$F(x)=\int_{x}^{x_{1}}f(x)dx$$
I ask which is the correct expression to F(x) when the variable x is placed in limit inferior.

 

[Mark44]

I saw that when you have a definite integral of a function f(x), you can to express the primitive, F(x), placing the variable x in limit superior of integral:
$$F(x)=\int_{x_{0}}^{x}f(x)dx$$
It's known... So I ask if F(x) can be equal to this too:
$$F(x)=\int_{x}^{x_{1}}f(x)dx$$

Probably not.
$$\int_x^{x_1}f(x)dx = -\int_{x_1}^x f(x)dx$$
Does that look the same as the first one you defined as F(x) above?

I ask which is the correct expression to F(x) when the variable x is placed in limit inferior.

Two things:
1. The integrals you wrote are functions of x. The dummy variable in the integral should be some other variable, such as t. In other words, your first integral probably should be written like this:
$$F(x) = \int_{x_0}^x f(t)dt$$
2. You should identify two different things with the same letter. In other words, the two integrals you wrote should not both be identified as F(x).