Editing Improper integral (section)

== Convergence of the integral ==
An improper integral converges if the limit defining it exists.  Thus for example one says that the improper integral
:<math>\lim_{t\to\infty}\int_a^t f(x)\ dx</math>
exists and is equal to ''L'' if the integrals under the limit exist for all sufficiently large ''t'', and the value of the limit is equal to ''L''.

It is also possible for an improper integral to diverge to infinity.  In that case, one may assign the value of ∞ (or −∞) to the integral. For instance 
:<math>\lim_{b\to\infty}\int_1^b \frac{dx}{x} = \infty.</math>
However, other improper integrals may simply diverge in no particular direction, such as
:<math>\lim_{b\to\infty}\int_1^b x\sin(x)\,dx,</math>
which does not exist, even as an [[extended real number]]. This is called divergence by oscillation.

A limitation of the technique of improper integration is that the limit must be taken with respect to one endpoint at a time.  Thus, for instance, an improper integral of the form

:<math>\int_{-\infty}^\infty f(x)\,dx</math>

can be defined by taking two separate limits; to which

:<math>\int_{-\infty}^\infty f(x)\,dx = \lim_{a\to -\infty} \lim_{b\to\infty} \int_a^bf(x)\,dx</math>

provided the double limit is finite. It can also be defined as a pair of distinct improper integrals of the first kind:

:<math>\lim_{a\to -\infty}\int_a^c f(x)\,dx + \lim_{b\to\infty} \int_c^b f(x)\,dx</math>

where ''c'' is any convenient point at which to start the integration. This definition also applies when one of these integrals is infinite, or both if they have the same sign.

An example of an improper integral where both endpoints are infinite is the [[Gaussian integral]] {{nowrap|<math display="inline">\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}</math>.}} An example which evaluates to infinity is {{nowrap|<math display="inline">\int_{-\infty}^\infty e^x\,dx</math>.}} But one cannot even define other integrals of this kind unambiguously, such as {{nowrap|<math display="inline">\int_{-\infty}^\infty x\,dx</math>,}} since the double limit is infinite and the two-integral method

:<math>\lim_{a\to -\infty}\int_a^cx\,dx + \lim_{b\to\infty} \int_c^b x\,dx</math>
yields an [[indeterminate form]], {{nowrap|<math>\infty-\infty</math>.}} In this case, one can however define an improper integral in the sense of [[Cauchy principal value]]:

:<math> \operatorname{p.v.} \int_{-\infty}^\infty x\,dx = \lim_{b\to\infty}\int_{-b}^b x\,dx = 0.</math>

The questions one must address in determining an improper integral are:

*Does the limit exist?
*Can the limit be computed?

The first question is an issue of [[mathematical analysis]]. The second one can be addressed by calculus techniques, but also in some cases by [[contour integration]], [[Fourier transform]]s and other more advanced methods.