The Improper Integral Calculator evaluates integrals that cannot be treated as ordinary finite-interval integrals because the interval is infinite, the integrand becomes unbounded, or both. These cases must be defined through limits.
What Makes an Integral Improper
An integral such as β«ββf(x)dx is improper because the upper limit is infinite. An integral can also be improper when f(x) becomes undefined or unbounded inside the interval. In either case, the calculation is interpreted as a limit of proper integrals.
Example With an Infinite Limit
For β«ββ1/xΒ² dx, replace the infinite endpoint with b and evaluate β«βα΅1/xΒ²dx. The result is 1β1/b. Taking the limit as b approaches infinity gives 1, so the improper integral converges.
Example of Divergence
For β«ββ1/x dx, the corresponding limit produces ln(b), which grows without bound as b increases. Therefore the integral diverges rather than having a finite value.
Using the Calculator
Enter the integrand and limits in the format supported by the tool. Pay attention to singular points inside a finite interval because the integral may need to be split into separate limits.
Do Not Ignore Convergence
A symbolic result is not enough for an improper integral. The limiting process must converge to a finite value for the integral to be considered convergent. When the calculator reports divergence, that result is mathematically meaningful rather than a calculation failure.