Is it mathematically correct to write the primitive of a function in this way : $F(x)=\int f(x)dx$
or should we absolutely change the variable name : as in $F(t)=\int f(x)dx$ ? (that is : not use the same variable inside the integral)
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1$\begingroup$ The first one is better form. Even better: $F(x) = \int f(x)dx + C$. For example, $$\cos x = \int \sin x \space dx + C.$$ The second one is more appropriate for definite integrals, for example $$F(t) = \int_a^t f(x)dx.$$ $\endgroup$– D_SCommented Mar 6, 2021 at 16:51
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$F(x)=\int f(x)dx$ is very common. So you will never convince everyone not to do it.
However $F(t)=\int f(x)dx$ is worse. The RHS does not reveal what the variable is.
Some peculiar people may write $F = \int f$, which is OK as far as variables are concerned.
Related: Do not write $F(x)=\int_0^x f(x)dx$. That has $x$ used for two different things in the same formula.
But it is OK to write $F(t)=\int_0^t f(x)dx$. This time the RHS does reveal what the variable is.
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$\begingroup$ Thank you for your kind explanations. $\endgroup$ Commented Mar 6, 2021 at 17:02