Possible Duplicate:
Why is $\int^\infty_{-\infty} \frac{x}{x^2+1} dx$ not zero?
I've always learned that the improper integral $\int\limits_{-\infty}^{\infty}f(x)dx$ only exists when the integrals $\int\limits_{-\infty}^af(x)dx$ and $\int\limits_a^{\infty}f(x)dx$ exists. Why is that?
If we have for example the function $f(x)=x^3$ then the integral $\int\limits_{-a}^{a}f(x)dx$ is defined for any a. However, $\int\limits_{-\infty}^{\infty}f(x)dx$ isn't defined. Why is that?