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8.8 Improper integrals Recall that in the study of definite integral (^) ∫ b
a
f (x) dx,
we assume that
- the interval [a, b] is finite
- the function f is bounded.
Such integrals are said to be proper. How about the area under the following curves?
This leads to improper integrals! Improper integrals
- Type I: the interval is infinite: [a, ∞), (−∞, b] or (−∞, ∞).
- Type II: the function f is unbounded. Idea: improper integrals are calculated as limits. Type I: Integral over infinite intervals.
- If f (x) is continuous on [a, ∞), then (^) ∫ ∞
a
f (x) dx = lim b→∞
∫ (^) b
a
f (x) dx.
- If f (x) is continuous on (−∞, b], then (^) ∫ b
−∞
f (x) dx = (^) a→−∞lim
∫ (^) b
a
f (x) dx.
- If f (x) is continuous on (−∞, ∞), then ∫ (^) ∞
−∞
f (x) dx =
∫ (^) a
−∞
f (x) dx +
a
f (x) dx
where a is any real number.
In each case, if the limit is finite, we say the improper integral converges and that the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges. Note that
−∞ f^ (x)^ dx^ converges if and only if both^
∫ (^) a −∞ f^ (x)^ dx^ and^
a f^ (x)^ dx^ converge. Example: Evaluate (^) ∫ (^) ∞
0
dx e^3 x
Then, (^) ∫ (^) ∞
a
f (x) dx and
a
g (x) dx
both converge or both diverge. Example: Does the integral (^) ∫ ∞
2
dx x
x^2 − 1 converge? Example:Does the integral (^) ∫ ∞
0
5 + sin
tan^2
x^3
− 4 ex+log x 5 )
e^3 x^
dx
converge? Type II Improper integral: integrals of unbounded function.
Important Example: Evaluate (^) ∫ 1
0
xp^
dx
For p ≤ 0 , the integral
0
1 xp^ dx^ is a proper integral. For^ p >^0 , it is an improper integral. In this case, we define ∫ (^1)
0
xp^
dx = lim a→0+
a
xp^
dx
c.f. the improper integral (^) ∫ (^) ∞
1
xp^
dx
- diverges for p ≤ 1.
- converges for p > 1;
Example: (^) ∫ 1
− 1
dx x^2 Example: (^) ∫ 1
− 1
dx x − 1 In general, (^) ∫ b
a
(x − a)p^
dx and
∫ (^) b
a
(x − b)p^
dx
converges if p < 1 and diverges if p ≥ 1. Example: (^) ∫ 3
0
dx (x − 1)^2 /^3
