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I know that this should not be done, but I was wondering if there is an intuitive reason why a function of x should not be in an integral limit.

In the same vein, suppose I had limits of an jntegral with respect to x, which were functions of a variable a. Now if I were to take the derivative of this integral with respect to a, I cannot pull the derivative inside of the integral sign without making some corrections (I have just read about partial derivatives and do not have a full understanding of them yet, but I know there are extra terms when you 'differentiate under the integral sign').

I think the answer underlying both of the above points stems from the same idea, which is why I grouped them together in one question.

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    There isn't any problem with having a function of x in an integral limit, proper practice is to then use a different letter for the integration variable to make it less ambiguous – 2017-01-13

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If $x$ is the variable of integration and a limit of integration, then it's doing double duty. When you integrate from $a$ to $b$, then you think of $x$ traveling from $a$ to $b$ and accumulating little bits of area at each point.

But if you integrate from $a$ to $x$, then $x$ is having an out-of-body experience. It's living at the endpoint, but also is transporting itself near $a$.