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Solution

Let $I = \int e^{x^2}\left(\frac{1}{x} - \frac{1}{2x^2}\right)dx$. Differentiate $e^{x^2}/x$: $\dfrac{d}{dx}\left(\dfrac{e^{x^2}}{x}\right) = e^{x^2}\left(2 - \dfrac{1}{x^2}\right)$. Thus, $I$ can be expressed as a part of $\dfrac{d}{dx}\left(\dfrac{e^{x^2}}{2x}\right)$, and on integration we get: $I = \dfrac{e^{x^2}(e^2 - 2)}{2} + C$.