Numerically approximate improper integrals with infinite limits or endpoint singularities. Enter \(f(x)\), the lower and upper limits, and the numerical resolution to calculate an approximation and check its stability.
Enter an expression such as 1/x^2, exp(-x), or 1/sqrt(x).
An improper integral is an integral that cannot be evaluated directly as an ordinary finite integral because either the integration interval is infinite or the function becomes unbounded or undefined at a point in the interval.
For example,
has an infinite upper limit. It is defined using a limit:
Similarly, an integral such as
is improper because the function becomes unbounded as \(x\to0^+\).
There are several common types of infinite-interval integrals.
Such integrals require particular care because convergence on both sides must be considered.
when the two improper integrals converge separately.
An integral can also be improper when the function is undefined or unbounded at an endpoint.
For example:
is improper because \(1/\sqrt{x}\) becomes unbounded at \(x=0\).
It is important to distinguish between the behavior of the function and the convergence of the integral. A function can become infinite at an endpoint while the area represented by the improper integral remains finite.
This is a convergent improper integral.
The exponential function decreases rapidly enough for the integral to converge.
Although the integrand becomes unbounded at zero, the improper integral is finite.
This integral diverges because its value grows without approaching a finite limit.
An improper integral converges if its defining limit exists and is finite.
It diverges if the limit does not exist or becomes infinite.
Numerical integration can provide useful evidence about the behavior of an integral, but numerical stability should not be confused with a mathematical proof of convergence.
For important theoretical problems, analytical convergence tests should be used in addition to numerical calculation.
The calculator supports:
sqrt(x)
abs(x)
sin(x),
cos(x),
tan(x)
exp(x)
ln(x) and log(x)
pi and e
Examples include
1/x^2,
exp(-x),
1/sqrt(x),
and
sin(x)/x.
For regular finite intervals, the calculator uses the composite Simpson rule.
Infinite intervals are transformed to finite numerical intervals, and endpoint singularities are handled using variable transformations before applying numerical integration.
The calculator also repeats the calculation at a lower numerical resolution to look for significant changes in the result.
An improper integral is an integral with an infinite limit of integration or a function that becomes unbounded or undefined at a point in the integration interval.
Improper integrals are defined using limits. For numerical calculation, the infinite interval or singular endpoint can be transformed into a finite numerical interval and then approximated.
Yes. Examples include integrals from a finite number to infinity, from negative infinity to a finite number, and from negative infinity to positive infinity.
Yes. For example, an integral may contain a function such as 1/sqrt(x) whose value is unbounded at an endpoint while the improper integral itself can still converge.
No. A numerical approximation can suggest stability, but convergence is a mathematical property that may require analytical analysis. This calculator therefore reports a numerical approximation and a basic stability check rather than a proof of convergence.
The calculator supports arithmetic operations, powers, x, pi, e, square root, absolute value, sine, cosine, tangent, exponential, natural logarithm and logarithm.