We say thatf isnonoscillatoryatb ._{D 1} if b _{D 1}/iff is defined onŒa; b/and
does not change sign on some subintervalŒa1; b/ofŒa; b/. Iff changes sign on every
such subinterval,f isoscillatoryatb . For a function that is locally integrable onŒa; b/
and nonoscillatory atb , convergence and absolute convergence ofR_{a}bf .x/ dx amount
to the same thing (Exercise3.4.16), so absolute convergence is not an interesting concept
in connection with such functions. However, an oscillatory function may be integrable,
but not absolutely integrable, onŒa; b/, as the next example shows. We then say thatf is
conditionallyintegrable onŒa; b/, and thatR_{a}bf .x/ dxconvergesconditionally.

### Example 3.4.14

We saw in Example3.4.13that the integralI.p/_{D}

Z 1

1 sinx

xp dx

is not absolutely convergent if0 < p_{}1. We will show that it converges conditionally for
these values ofp.

Integration by parts yields
Z c
1
sinx
xp dxD
cosc
cp Ccos1 p
Z c
1
cosx
xpC1dx: (3.4.8)
Since _{ˇ}
ˇ
ˇcosx
xpC1
ˇ
ˇ
ˇ 1
xpC1

andR_{1}1x p 1_{dx <}_{1}_{if}_{p > 0}_{, Theorem}_{3.4.6}_{implies that}_{x} p 1_{cos}_{x} _{is absolutely}
integrableŒ1;_{1}/ifp > 0. Therefore, Theorem3.4.9implies thatx p 1_{cos}_{x}_{is integrable}

Œ1;_{1}/ifp > 0. Lettingc_{! 1}in (3.4.8), we find thatI.p/converges, and

I.p/_{D}cos1 p

Z 1

1 cosx

xpC1dx if p > 0:
This and (3.4.7) imply thatI.p/converges conditionally if0 < p_{}1.

The method used in Example3.4.14is a special case of the following test for convergence of improper integrals.

### Theorem 3.4.10 (Dirichlet’s Test)

Suppose thatf is continuous and its an- tiderivativeF .x/_{D}R

_{a}xf .t / dt is bounded onŒa; b/:Letg0 be absolutely integrable on

Œa; b/;and suppose that

lim

x!b g.x/D0: (3.4.9)

ThenR_{a}bf .x/g.x/ dxconverges:

### Proof

The continuous functionfgis locally integrable onŒa; b/. Integration by parts yields Z c a f .x/g.x/ dx_{D}F .c/g.c/ Z c a F .x/g0.x/ dx; a

_{}c < b: (3.4.10) Theorem3.4.6implies that the integral on the right converges absolutely asc

_{!}b , since Rb

a jg0.x/jdx <1by assumption, and

jF .x/g0.x/_{j }M_{j}g0.x/_{j};

whereM is an upper bound for_{j}F_{j}onŒa; b/. Moreover, (3.4.9) and the boundedness ofF
imply that limc!b F .c/g.c/D0. Lettingc!b in (3.4.10) yields

Z b
a
f .x/g.x/ dx _{D}
Z b
a
F .x/g0.x/ dx;
where the integral on the right converges absolutely.

Dirichlet’s test is useful only iff is oscillatory atb , since it can be shown that iff is
nonoscillatory atb andF is bounded onŒa; b/, thenR_{a}b_{j}f .x/g.x/_{j}dx <_{1}if onlygis
locally integrable and bounded onŒa; b/(Exercise3.4.14).

### Example 3.4.15

Dirichlet’s test can also be used to show that certain integrals di-verge. For example, _{Z}

1

1

xqsinx dx

diverges ifq > 0, but none of the other tests that we have studied so far implies this. It
is not enough to argue that the integrand does not approach zero asx _{! 1}(a common
mistake), since this does not imply divergence (Exercise4.4.31). To see that the integral
diverges, we observe that if it converged for someq > 0, thenF .x/ _{D} R_{1}xxqsinx dx
would be bounded onŒ1;_{1}/, and we could let

f .x/_{D}xqsinx and g.x/_{D}x q
in Theorem3.4.10and conclude that

Z 1

1

sinx dx

The method used in Example3.4.15is a special case of the following test for divergence of improper integrals.

### Theorem 3.4.11

Suppose thatuis continuous onŒa; b/andR_{a}bu.x/ dxdiverges:Let

vbe positive and differentiable onŒa; b/;and suppose thatlimx!b v.x/D 1andv0=v2
is absolutely integrable onŒa; b/:ThenR_{a}bu.x/v.x/ dxdiverges:

### Proof

The proof is by contradiction. Letf_{D}uv andg

_{D}1=v, and suppose that Rb

a u.x/v.x/ dxconverges. Thenf has the bounded antiderivativeF .x/D

Rx

a u.t /v.t / dt onŒa; b/, limx!1g.x/D0andg0D v0=v2is absolutely integrable onŒa; b/. Therefore,

Theorem3.4.10implies thatR_{a}bu.x/ dxconverges, a contradiction.

If Dirichlet’s test shows thatR_{a}bf .x/g.x/ dxconverges, there remains the question of
whether it converges absolutely or conditionally. The next theorem sometimes answers this
question. Its proof can be modeled after the method of Example3.4.13(Exercise3.4.17).
The idea of an infinite sequence, which we will discuss in Section 4.1, enters into the
statement of this theorem. We assume that you recall the concept sufficiently well from
calculus to understand the meaning of the theorem.

### Theorem 3.4.12

Suppose thatgis monotonic onŒa; b/andR_{a}bg.x/ dx

_{D 1}:Letf

be locally integrable onŒa; b/and

Z xjC1 xj

jf .x/_{j}dx_{}; j _{}0;

for some positive;where_{f}xjgis an increasing infinite sequence of points inŒa; b/such
thatlimj!1xj DbandxjC1 xj M; j 0;for someM:Then

Z b a j

f .x/g.x/_{j}dx _{D 1}: