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1.49 Integrals over the real line

We can likewise define

\int_{-\infty}^{b}f(x)\,dx=\lim_{A\rightarrow-\infty}\int_{A}^{b}f(x)\,dx

where this limit exists. The improper integral of $f$ over the real line is defined to be

\int_{-\infty}^{\infty}f(x)\,dx=\int_{0}^{\infty}f(x)\,dx+\int_{-\infty}^{0}f(% x)\,dx

when both of these improper integrals on the right-hand side exist.

Notation. Formulae such as $\infty-\infty$ or $2\infty/\infty$ are nonsense. We must make sure that infinite areas do not cancel out.