Page 315 - Schaum's Outline of Theory and Problems of Advanced Calculus

Improper Integrals

DEFINITION OF AN IMPROPER INTEGRAL
The functions that generate the Riemann integrals of Chapter 6 are continuous on closed intervals.
Thus, the functions are bounded and the intervals are finite. Integrals of functions with these char-
acteristics are called proper integrals. When one or more of these restrictions is relaxed, the integrals are
said to be improper. Categories of improper integrals are established below.
ð b
The integral f ðxÞ dx is called an improper integral if
a
1. a ¼ 1 or b ¼1 or both, i.e., one or both integration limits is infinite,
2. f ðxÞ is unbounded at one or more points of a @ x @ b. Such points are called singularities of
f ðxÞ.
Integrals corresponding to (1) and (2) are called improper integrals of the first and second kinds,
respectively. Integrals with both conditions (1) and (2) are called improper integrals of the third kind.
ð
1
2
EXAMPLE 1. sin x dx is an improper integral of the first kind.
0
ð 4 dx
EXAMPLE 2. is an improper integral of the second kind.
0 x 3
e
ð 1 x
EXAMPLE 3. p ffiffiffi dx is an improper integral of the third kind.
0 x
ð 1 sin x sin x
EXAMPLE 4. dx is a proper integral since lim ¼ 1.
0 x x!0þ x
IMPROPER INTEGRALS OF THE FIRST KIND (Unbounded Intervals)
ð x ð a
If f is an integrable on the appropriate domains, then the indefinite integrals f ðtÞ dt and f ðtÞ dt
a x
(with variable upper and lower limits, respectively) are functions. Through them we define three forms
of the improper integral of the first kind.
Definition ð ð
1 x
(a)If f is integrable on a @ x < 1, then f ðxÞ dx ¼ lim f ðtÞ dt.
a x!1 a
ð a ð a
(b)If f is integrable on 1 < x @ a, then f ðxÞ dx ¼ lim f ðtÞ dt:
x! 1 x
1
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