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This document, from a lecture at the university of houston, covers the concept of improper integrals, the comparison test, unbounded intervals, and unbounded functions. It includes examples and properties of improper integrals, and defines the integral of unbounded functions.
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Improper Integrals Comparison Test
Section 10.7 Improper Integrals
Jiwen He Department of Mathematics, University of Houston [email protected] http://math.uh.edu/∼jiwenhe/Math
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
1
x^2 dx^ =?,
0
x^2 dx^ =?
Known:
∫ (^) b
a
f (x) dx =
∫ (^) b
a
x^2
dx =
x
]b
a
a
b
, 0 < a < b,
the interval of integration [a, b], 0 < a < b, is bounded, the function being integrated f (x) = (^) x^12 is bounded over [a, b].
By a limit process, we can extend the integration process to unbounded intervals (e.g., [1 ∫ , ∞)): ∞ 1
x^2 dx^ =^ blim→∞
∫ (^) b
1
x^2 dx^ =^ blim→∞
b
unbounded functions (e.g., as ∫ x → 0 +, f (x) = (^) x^12 → ∞): 1 0
x^2 dx^ =^ alim→ 0 +
a
x^2 dx^ =^ alim→ 0 +
a −^
Let f be continuous on [a, ∞). We define ∫ ∞ a
f (x) dx = lim b→∞
∫ (^) b
a
f (x) dx
The improper integral converges if the limit exists. The improper integral diverges if the limit doesn’t exist.
If (^) blim→∞ f (x) 6 = 0, then
a
f (x) dx diverges.
If f is continuous on (−∞, b],
∫ (^) b
−∞
f (x) dx = (^) a→−∞lim
∫ (^) b
a
f (x) dx.
If f is cont. on (−∞, ∞),
−∞
f (x) dx =
−∞
f (x) dx +
0
f (x) dx
Let f be continuous on [a, ∞). We define ∫ ∞ a
f (x) dx = lim b→∞
∫ (^) b
a
f (x) dx
The improper integral converges if the limit exists. The improper integral diverges if the limit doesn’t exist.
If (^) blim→∞ f (x) 6 = 0, then
a
f (x) dx diverges.
If f is continuous on (−∞, b],
∫ (^) b
−∞
f (x) dx = (^) a→−∞lim
∫ (^) b
a
f (x) dx.
If f is cont. on (−∞, ∞),
−∞
f (x) dx =
−∞
f (x) dx +
0
f (x) dx
