The short answer:
An integral is improper if the
integrand has a discontinuity. That is, when
the function inside the integral does something bad. Gavin's general
rules for figuring out when this might be the case:
- Whenever there is division by zero. Look out for powers of
x in the denominator of fractions.
- This rule means that we have to be careful whenever there's a
negative exponent, too.
- Logarithms (ln(x) and log(x), for example) are not
defined for x less than or equal to zero, and are
therefore discontinuous whenever their argument is zero.
- Square roots (x1/2)
are not defined for x less than zero. However, they
are defined for x=0, so there isn't a
discontinuity there.
- Tangent is sin(x)/cos(x), so it's discontinuous
whenever cos(x) is zero, that is, for odd multiples of pi/2.
Cotangent is cos(x)/sin(x), so it's discontinuous
where sin(x) is zero, or any multiple of pi. Similarly,
sec(x) is 1/cos(x) and csc(x) is 1/sin(x).
It's probably worth committing these to memory.