Why is another variable than "x" , namely, variable " t" , substituted for x in the expression " integral from a to x of f(t)dt"?

Question

The documents I have consulted quickly pass on the question of knowing why an indefinite integral of a function f is not written as " integral from a to x of f(x)dx" but rather as " integral from a to x of f(t)dt".

What confusion or non-sense would it produce to let the variable x remain in the expression?

Why is this variable change necessary in the case of definite integral, and not in the case of indefinite integral?

Answer 1

Because a variable is supposed yo have one specific meaning in an equation (or a collection of them that forms some closed whole). The $t$ as variable over which you integrate is one meaning (it takes the values between the limits, if you recall what integration means), the limit is fixed.

Answer 2

In definite integration, the variable of integration is basically a dummy variable used for comprehensional purposes. To put that into context, suppose we had: $$\int_a^bf(x)\ dx$$ If we then performed a substitution, say $e^x$, then using the variable $x$ again would be confusing. It just wouldn't make sense to have $x=e^x$. In fact, this specific case would make the expression impossible to solve! Rather, we could use another variable, like $t$. Setting $t=e^x$ makes so much more sense.