Divergent integral

A concept opposite to that of a convergent integral (see also Singular integral). For example, if a function $ f $ is defined on a bounded or unbounded interval $ [ a, b) $, $ - \infty \leq a \leq b \leq \infty $, if for each $ \eta \in [ a, b) $ it is integrable on $ [ a, \eta ] $ and if there is no finite limit

$$ \lim\limits _ {\eta \rightarrow b } \ \int\limits _ { a } ^ \eta f ( x) dx, $$

then one says that the integral $ \int _ {a} ^ {b} f ( x) dx $ diverges. In the case that

$$ \lim\limits _ {\eta \rightarrow b } \ \int\limits _ { a } ^ \eta f ( x) dx = \ + \infty \ \textrm{ or } \ \ - \infty , $$

one says that the divergent integral $ \int _ {a} ^ {b} f ( x) dx $ is equal to $ + \infty $ or $ - \infty $, respectively.