1. State the problem: Compute the indefinite integral $\int x e^{x^{2}-3}\,dx$.\n2. Method and formula: We use $u$-substitution because the integrand is $x$ times a function of $x^{2}$.\n3. Choose the substitution: Let $u = x^{2}-3$.\n4. Differentiate to relate $du$ and $dx$: Then $du = 2x\,dx$.\n$$du = 2x\,dx$$\nIntermediate cancellation when solving for $x\,dx$:\n$$\frac{du}{\cancel{2}} = x\,dx$$\n5. Substitute into the integral: Replace $x\,dx$ with $\dfrac{du}{2}$ and $x^{2}-3$ with $u$.\n$$\int x e^{x^{2}-3}\,dx = \int e^{u} \frac{du}{2}$$\nShow factoring out the constant with cancellation:\n$$\int e^{u} \frac{du}{2} = \frac{1}{\cancel{2}}\int e^{u}\,du$$\n6. Integrate with respect to $u$: The antiderivative of $e^{u}$ is $e^{u}$, so we get\n$$= \frac{1}{2}e^{u} + C$$\n7. Back-substitute $u = x^{2}-3$ to return to the original variable and state the final answer:\n$$\boxed{\frac{1}{2}e^{x^{2}-3} + C}$$\n