probability – Find the mean of $Y=frac{1}{A}(e^{A}-1)$ where $Asim N(0,sigma^2)$


A sub-question of an exercise from my probability class is as follows:

Let $Asim N(0,sigma^2)$ and $X(t)=e^{At}$. Find the mean of $Y=int_0^1X(t)dt$.

I figured that we have $Y=frac{1}{A}(e^{A}-1)cdotunicode{x1D7D9}_{{Aneq 0}}$ so that the mean is equal to

begin{align} mathbb{E}(Y) & = frac{1}{sqrt{2pisigma^2}}int_{-infty}^inftyfrac{e^x-1}{x}exp{(-frac{x^2}{2sigma^2})}dx.
end{align}

I do, however, have not the slightest idea as to how I should proceed from here. I am doubting whether I am even approaching the question in the right way.