Improper integrals

  • Improper integrals are a type of definite integral where either or both of the limits are infinite, or where the integrand becomes infinite at one or more points in the range of integration.

  • For an integral with infinite limits such as ∫ from a to ∞ f(x) dx, we say the integral converges if the limit as t approaches ∞ of ∫ from a to t f(x) dx exists. If the limit does not exist, we say the integral diverges.

  • For an integral where the integrand becomes infinite, also known as a Type II improper integral, we say the integral converges if the limit as t approaches the point of singularity of the integral from a to t of f(x) dx exists. If the limit does not exist, we say the integral diverges.

  • Techniques for evaluating infinite integrals may include substitution, integration by parts, or using known results of specific integral forms.

  • To evaluate an integral with an infinite discontinuity, split the integral into two at the point of the discontinuity and take the limit separately on each side.

  • In total there are four types of integrals that qualify as improper: (1) Infinite limits of integration, (2) Discontinuous integrand within the interval of integration, (3) Discontinuous integrand at a finite endpoint of the interval of integration, (4) Infinite integrand at a finite point within the interval of integration.

  • The comparison test for improper integrals can be used to determine convergence or divergence. If 0≤f(x)≤g(x) for a ≤ x, then if ∫ g(x) dx converges so does ∫ f(x) dx, and if ∫ f(x) dx diverges, so does ∫ g(x) dx.

  • Remember to always check whether an integral is improper. It would be a common mistake to assume an integral is proper because it has finite limits of integration, without checking for points of discontinuity of the integrand within the interval of integration.

  • Lastly, keep in mind that some improper integrals, despite being divergent, can be assigned a finite value using a process known as regularization. However, this is beyond the scope of the Core Pure Mathematics 2 content and is usually covered in higher level courses.