How to write correctly when performing integral calculations?

Question

I have heard that when integrating, another variable must be used. But I'm not clear when to use it, and why this can lead to confusion. I do not know if it's just a good practice or it's a rule.

What I want is to express the calculations correctly, but I do not know when to change the name of the variable. How should it be done?

For example

If I have an equation like this: $f(x)=x^2+x+1$

And I want to calculate the following integrals:

1. $\displaystyle \int f(x)\,dx$
2. $\displaystyle \int_{-\infty}^{+\infty} f(x)\,dx$
3. $\displaystyle \int_{-\infty}^{0} f(x)\,dx$
4. $\displaystyle \int_{0}^{1} f(x)\,dx$
5. $\displaystyle \int_{1}^{+\infty} f(x)\,dx$

What is the correct notation that I should use to solve them?

Thank you very much.

Answer 1

There has been much debate over whether the following is acceptable:

$$\int_0^x dx \, f(x) $$

To me, this is horribly confusing, as confusing as this:

$$\sum_{k=0}^k a_k $$

So I prefer

$$\int_0^x dt \, f(t) $$

This comes in handy if, say, you want to do a substitution like $t=x u$:

$$x \int_0^1 du \, f(x u) $$

It also comes in handy when, say, squaring integrals, like

$$\left [ \int_0^x dt \, f(t) \right ]^2 = \int_0^x dt \, f(t) \, \int_0^x dt' \, f(t') $$

Answer 2

It's just a good habit since $F(x) = \int_0^x f(x) dx$ might cause confusion whereas $F(x) = \int_0^x f(t) dt$ probably won't.