Improper integrals

We de¯ned
R b
a f(t)dt under the conditions that f is de¯ned and bounded on the bounded interval
[a; b]. In this lecture, we will extend the theory of integration to bounded functions de¯ned on
unbounded intervals and also to unbounded functions de¯ned on bounded or unbounded intervals.
Improper integral of the ¯rst kind: Suppose f is (Riemann) integrable on [a; x] for all x > a,
i.e.,
R x
a f(t)dt exists for all x > a. If lim
x!1
R x
a f(t)dt = L for some L 2 R; then we say that
the improper integral (of the ¯rst kind)
R 1
a f(t)dt converges to L and we write
R 1
a f(t)dt = L.
Otherwise, we say that the improper integral
R 1
a f(t)dt diverges.
Observe that the de¯nition of convergence of improper integrals is similar to the one given for
series. For example,
R x
a f(t)dt; x > a is analogous to the partial sum of a series.
Examples : 1. The improper integral
R 1
1
1
t2 dt converges, because,
R x
1
1
t2 dt = 1? 1
x ! 1 as x ! 1:
On the other hand,
R 1
1
1
t dt diverges because limx!1
R x
1
1
t dt = limx!1 log x. In fact, one can show
that
R 1
1
1
tp dt converges to 1
p?1 for p > 1 and diverges for p · 1.
2. Consider
R 1
0 te?t2
dt: We will use substitution in this example. Note that
R x
0 te?t2
dt = 1
2
R x2
0 e?sds = 1
2(1 ? e?x2) ! 1
2 as x ! 1:
3. The integral
R 1
0 sintdt diverges, because,
R x
0 sintdt = 1 ? cosx:
We now derive some convergence tests for improper integrals. These tests are similar to those
used for series. We do not present the proofs of the following three results as they are similar to
the proofs of the corresponding results for series.