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Material Type: Notes; Class: Calculus II; Subject: Mathematics; University: Pellissippi State Technical Community College; Term: Unknown 1989;
Typology: Study notes
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Improper Integrals Section 5. The definition of the definite integral requires the interval over which we are integrating to be finite. The Evaluation Theorem further requires the integrand to be a continuous function over the interval we are integrating. Integrals with infinite intervals of integration and integrals with integrands that have an infinite discontinuity (i.e vertical asymptote) at some point on the interval of integration are called improper integrals.
INTEGRALS WITH INFINITE INTERVALS OF INTEGRATION ( TYPE 1 )
t fx dx (provided the limit exists)
b
b fx dx (provided the limit exists)
If one of the above limits exist, we say the corresponding improper integral CONVERGES. If the limit does not exist (or is infinite), we say the improper integral DIVERGES.
−
−
a
a
fx dx provided both the integrals on the left converge.
Examples:
x^2
dx
x dx
x p^ dx
0 e x^ dx
1
1 − xe−x^ dx
(^) e x 1 e^2 x^
dx
-3 -2 -1 1 2 3
x
y
Graph of y e^
x 1 e^2 x
b fx dx lim t→b−^
a
t fx dx
b fx dx lim t→a^
b fx dx
The improper integral is called CONVERGENT if the corresponding limit exists. If the limit does not exist (or is infinite) then the improper integral is DIVERGENT.
a
b
a
c
c
b fx dx
provided these last two integrals are convergent.
Examples:
(^1) dx (^3) x
0
(^2) dx x^3
(^2) dx x^3
/ sec 2 x dx
1 ln x dx
COMPARISON TEST FOR IMPROPER INTEGRALS ( Type 1 Integrals ) Let f and g be continuous functions with fx ≥ gx ≥ 0 for x ≥ a.
gx dx converges.
fx dx diverges.
Example:
(^1) e x x dx^ is divergent since
1 e x x ^
x dx^ is divergent.
1
x^2 5
dx is convergent since 1 x^2
x^2 5
1
x^2
dx is convergent.