Interchanging from a function defined by an integral to indefinite integral

Question

This question might be trivial, but I have the following:

$$\operatorname{E}_n(x) = \int_{0}^{x} e^{-t^2}t^n \, dt$$

If I have $\operatorname{E}_n(x)$ as a function defined as an integral, it is correct to say

$$\operatorname{E}_n = \int e^{-x^2}x^n \, dx ?$$

Answer 1

$\int e^{-x^2}x^n \, dx$ denotes the set of all antiderivatives of $e^{-x^2}x^n$.

$E_n(x) = \int_{0}^{x} e^{-t^2}t^n \, dt$ is a special antiderivative of $e^{-x^2}x^n$ because of $E_n(0)=0.$

Answer 2

As the name indicates, an indefinite integral is not completely defined, it represents a family of functions. So, no it is not really correct.

To some extent one might admit the equivalence

$$\int f(t)\,dt=\int_a^x f(t) dt$$ where $a$ is unspecified.