There are two kinds of improper integral. The first kind is one in which the interval of integration is infinite as, for example, inIt is said that this integral exists, and that its value is l, if the value of the integral from a to X tends to a limit l as X → ∞. For example,and, as X → ∞, the right-hand side tends to 1. SoA similar definition can be given for the improper integral from −∞ to a. If both the integralsexist and have values l1 and l2, then it is said that the integral from −∞ to ∞ exists and that its value is l1+l2.
The second kind of improper integral is one in which the function becomes infinite at some point. Suppose first that the function becomes infinite at one of the limits of integration as, for example, inThe function 1/√x is not bounded on the closed interval [0, 1], so, in the normal way, this integral is not defined. However, the function is bounded on the interval [δ, 1], where 0 < δ < 1, andAs δ → 0, the right-hand side tends to the limit 2. So the integral above, from 0 to 1, is taken, by definition, to be equal to 2; in the same way, any such integral can be given a value equal to the appropriate limit, if it exists. A similar definition can be made for an integral in which the function becomes infinite at the upper limit.
Finally, an integral in which the function becomes infinite at a point between the limits is dealt with as follows. It is written as the sum of two integrals, where the function becomes infinite at the upper limit of the first and at the lower limit of the second. If both these integrals exist, the original integral is said to exist and, in this way, its value can be obtained.