Calculus 141, section 8.7a Improper Integrals
notes by Tim Pilachowski
The definite integrals encountered so far have been proper, i.e. they have been evaluated over a finite interval
on which the function is continuous. Improper integrals take one of two forms: a) the boundaries of integration
are unbounded (i.e. they go to ∞, – ∞, or both), or b) the integrand is unbounded on the interval of integration.
Think about the limits we’ve done: some go to a specific finite value, others go to infinity, still others do not
exist. We will, in fact, use limits to evaluate improper integrals. The basic method is to rewrite the integral as a
limit —
() ()
∫∫ ∞→
∞=b
a
b
adxxfdxxf lim , with associated versions for integrals involving –∞ and denominators
approaching 0.
Improper integrals that have a numeric value are said to be convergent. The rest are said to be divergent.
Example A: Does ∫∞
12
x
dx converge? Answer: converges to 1
Example A extended: Does ∫∞
02
x
dx converge? Answer: diverges
Example B: Does ∫∞
∞− +1
2
x
dx converge? Answer: converges to π
Example C: Does ∫∞
1x
dx converge? Answer: diverges
