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Integration of Improper Integrals
INTEGRATION OF IMPROPER INTEGRALS
Objectives: Evaluate integrals on infinite intervals and intervals which have infinite
discontinuities
Improper integrals
- May be defined on an infinite interval
- May have an infinite discontinuity
- Are used in probability distributions
The improper integrals and are said to be convergent if the a
f(x)dx
∞
b f(x)dx
corresponding [finite] limit exists and divergent if the [finite] limit does not exist.
Type I: Infinite intervals
t t
1 2 1
dx 1
x x^ t
- The unbounded region extends indefinitely in a horizontal direction
- Note that A(t) < 1 no matter how large t is chosen
t t
lim A(t) lim → ∞ → ∞
t
- Area of shaded region approaches 1 as t → ∞
t
1 2 1 2 t
dx lim dx 1
x x
∞
→ ∞
- The integral is convergent
Integrate
(^0 ) dx −∞ (^) 2x − 5
HINT:
0 0
t t
1 1 dx lim dx −∞ (^) 2x 5 →∞ 2x 5
=
Integration of Improper Integrals
Type 1 improper integral
- If exists for every number then
t
a
f(x)dx
t ≥ a, a
f(x)dx
∞
t
t a
lim f(x)dx, → ∞
provided this [finite] limit exists
- If exists for every number t b, then
b
t
f(x)dx
≤
b f(x)dx
b
t t
lim f(x)dx, → − ∞
provided this [finite] limit exists
- The improper integrals and are said to be convergent if the a
f(x)dx
∞
b f(x)dx
corresponding [finite] limit exists and divergent if the [finite] limit does not exist.
- If both and are convergent, then we define a
f(x)dx
∞
a f(x)dx
f(x)dx^ =^ +^ for any real number a.
∞
a f(x)dx
∫ − ∞ a
f(x)dx
∞
Is convergent or divergent? 1
dx x
∞
- = (ln t - ln 1) = (ln t) = t
lim → ∞
t
1
dx x
t
lim → ∞
t 1
[ ln | x |] = t
lim → ∞ t
lim → ∞
∞
- Therefore, is divergent. 1
dx x
∞
- Compare this result to which converges 1 2
dx
x
∞
- In general, is convergent if p > 1 and divergent if p 1 1 p
dx x
∞
≤
Type 2: Discontinuous integrands
- f is continuous on a finite interval [a, b)
- The unbounded region is infinite in a vertical direction
Type 2 improper integrals
- If f is continuous on [a, b) and is discontinuous at b, then =
b
a
f(x)dx
t b
lim − →
t
a
f(x)dx,
if this [finite] limit exists
- If f is continuous on [a, b) and is discontinuous at a, then =
b
a
f(x)dx
t a
lim
→
b
t
f(x)dx,
if this [finite] limit exists
