Final answer:
An improper integral is identified as such when it involves infinite limits of integration or an unbounded integrand within the interval of integration.
Step-by-step explanation:
An improper integral is an integral where either the interval of integration is infinite or the function being integrated becomes infinite at one or more points within the interval of integration.
These types of integrals require special techniques to evaluate because traditional methods of integration assume the interval is finite and the function is well-behaved (finite) throughout that interval.
To decide if an integral is improper, you would look for the following indicators:
- The limits of integration include infinity or negative infinity.
- The function has vertical asymptotes (becomes infinite) within the interval.
For example, the integral of 1/x from 1 to infinity is improper because it extends to infinity. Similarly, the integral of 1/((x-1)^2) from 0 to 2 is improper because the integrand has a vertical asymptote at x=1 within the interval of integration. Both of these cases require the use of limits to define the integrals.