1. Improper Integrals I

2. Improper Integrals Definition An integral is improper if either:
the interval of integration is infinitely long or
if the function has singularities in the interval of integration (or both).

3. Improper Integrals
Improper integrals cannot be defined as limits of Riemann sums. Neither can one approximate them numerically using methods based on evaluating Riemann sums.

4. IMPROPER INTEGRALS Examples1
The integral is improper because the interval of integration is infinitely long.

5. IMPROPER INTEGRALS Examples 2
is improper because the integrand has a singularity.

6. IMPROPER INTEGRALS Examples 3
is improper because the integrand has a singularity and the interval of integration is infinitely long.

7. Improper Integrals Definition
Assume that the function f takes finite values on the interval [a, ∞).
If the limit exists and is finite, the improper integral converges, and

8. Improper Integrals
Example
Hence the integral converges.

9. Improper Integrals Definition
Assume that the function f takes finite values on the interval [a, ∞).
If the limit does not exists or is not finite, the improper integral diverges

10. Improper Integrals
Example
Hence the integral diverges.

11. Improper Integrals Definition
Assume that the function f has a singularity at x = a.
If the limit exists and is finite, the improper integral converges, and

12. Improper Integrals
Example
Hence the integral converges.

13. Improper Integrals Definition
Assume that the function f has a singularity at x = a.
If the limit does not exist or is not finite, the improper integral diverges.

14. Improper Integrals
Example
Hence the integral diverges.

15. Improper Integrals Definition
If the function f has a singularity at a point c, a < c < b, then the improper integral converges if and only if both improper integrals and converge.
In this case

16. Improper Integrals
Example
Hence the integral converges.

17. Improper Integrals Definition
If the function f has a singularity at a point c, a < c < b, then the improper integral diverges if either or diverges.

18. Improper Integrals Example
Neither limits exists. The integral diverges.

19. Improper Integrals Warning The integral diverges.
Trying to compute that integral by the Fundamental Theorem of Calculus, one gets
This is an incorrect computation.

20. Summary
An integral is improper if either: the interval of integration is infinitely long or if the function has singularities in the interval of integration (or both). Such integrals cannot be defined as limits of Riemann sums. They must be defined as limits of integrals over finite intervals where the function takes only finite values.