Integrals with infinite limits or singularities; converge or diverge depending on decay rate.
Improper integrals have infinite limits or unbounded integrands and are evaluated as limits. ∫₁^∞ x⁻ᵖ dx converges only if p>1. They arise in Laplace and Fourier transforms and probability densities.
∫₀^∞ e^{-st}dt = 1/s for s>0.
Improper integrals decide whether infinite ranges or singular points give a finite answer, which matters for total energy and heat flux.
Integrals with infinite limits or singularities; converge or diverge depending on decay rate.
Improper integrals decide whether infinite ranges or singular points give a finite answer, which matters for total energy and heat flux.
Compare the integrand with a known function, such as one over x to a power. Convergence depends on how fast the function decays.
A point where the integrand becomes infinite. It can be integrable, as with one over the square root of x, or not, depending on its strength.
Typical applications include checking whether the energy of a pulse is finite, evaluating the total charge in a long-tailed distribution and testing whether a stress singularity at a crack tip is integrable.
Study "Centroids and moments" first (topic 37 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Replace ∞ with t then take limit. Comparison test judges convergence. Singularities handled at the endpoint.
∫₀^∞ e^{-st}dt = 1/s for s>0.
The next topic is "Sequences convergence tests", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 38 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
Want to practise? Open the interactive course for quick checks, quizzes, simulations and progress tracking.