Improper Integral
• The definite integral
• a or b infinite,
• f (x) has one or more points of discontinuity in the
interval [a, b].
• If f (x) is a
continuous function on [a, ∞), then
Figure
• If f (x) is a
continuous function on ( – ∞, b], then
Figure
Note: The improper integrals in 1071, 1072 are convergent if the limits exist and are finite;
otherwise the integrals are divergent.
Figure
If for some real number c, both of the integrals
in the right side are convergent, then the integral
• Comparison Therems
Let f (x) and g
(x) be continuous functions on the closed interval [a, ∞). Suppose that 0 ≤ g (x) ≤ f
(x) for all x in [a, ∞).
• Absolute Convergence
If
• Discontinuous Integrand
Let f (x) be a function which is
continuous on the interval [a, b) but discontinuous at x = b.
Then
Figure
• Let f (x) be
a continuous function for all real numbers x in the interval [a, b]
except for some point c in (a, b). Then
Figure
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