We break up $\int^\infty\_{-\infty} \dfrac{x}{x^2+1} dx$ into:

$$\lim\_{t\to -\infty} \int^0\_t \dfrac{x}{x^2+1} dx + \lim\_{t\to \infty} \int^t\_0 \dfrac{x}{x^2+1} dx$$

So, evaluated, this gives;

$$\lim\_{t \to \infty} \left(\frac{1}{2} \ln (1+t^2)\right) - \lim\_{t \to -\infty} \left(\frac{1}{2} \ln (1+t^2)\right)$$

But those two terms are essentially identical! It should be zero! Plus, the integrand is an odd function, so why is this undefined? Is it just an unspoken rule that once you encounter $\infty$ in a mathematical expression you should stop evaluating immediately?