Solve integral x⁴ exponential^x/2

let's tackle this integral.

Evaluate $\int x^4 e^{x^2/2} dx$.

This integral, $\int x^4 e^{x^2/2} dx$, is a non-elementary integral. This means it cannot be expressed in terms of elementary functions (polynomials, exponentials, logarithms, trigonometric functions, and their inverses) using a finite number of operations.

Integrals of the form $\int x^n e^{ax^k} dx$ are generally non-elementary unless $k$ is a divisor of $n+1$. In this case, $n=4$ and $k=2$. Since $n+1=5$ and 2 does not divide 5, the integral is non-elementary.

Therefore, this integral cannot be solved using standard elementary calculus techniques.

The final answer is $\boxed{\text{This integral cannot be expressed in terms of elementary functions.}}$.