\( \int e ^ { \frac { - 3 x ^ { 2 } } { 2 } } \cdot x d x \)

To solve the integral \( \int e^{-\frac{3x^2}{2}} \cdot x \, dx \), we can use the technique of substitution. Let's set \( u = -\frac{3x^2}{2} \), which implies that \( du = -3x \, dx \) or \( dx = -\frac{du}{3x} \). When substituting, we can see that the integral transforms as follows: By substituting, we have: \[ \int e^{-u} \cdot \left(-\frac{1}{3}\right) e^{\frac{u}{3}} du. \] This integration might be more complex, but the simpler method is to recognize that this integral can be evaluated using Gaussian integrals. Specifically, the given form closely resembles the form used in finding the moments of a Gaussian distribution. Thus, the solution leads to a result involving the error function. In summary, while the specifics of handling such integrals can be tricky, the magic often lies in recognizing patterns and how they relate to well-known results. The final answer to this integral is given by: \[ -\frac{1}{3} e^{-\frac{3x^2}{2}} + C. \]