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8.8 Improper Integrals
Math 6B
Calculus II
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Type 1: Infinite Integrals
Definition of an Improper Integral of Type 1
provided this limit exists (as a finite number).
a) If ( ) exists for every number , thena
f x dx t a
( ) lim ( )
t
ta a
f x dx f x dx
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Type 1: Infinite Integrals
provided this limit exists (as a finite number).
b) If ( ) exists for every number , thenb
t
f x dx t b( ) lim ( )
b b
tt
f x dx f x dx
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Type 1: Infinite Integrals
The improper integrals are called convergent if the corresponding limit exists and divergent if the limit does not exist.
( ) and ( )b
a
f x dx g x dx
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Type 1: Infinite Integrals
-
c) If ( ) and ( ) are convergent, then we definea
a
f x dx f x dx
( ) ( ) ( )a
a
f x dx f x dx f x dx
In part c) any real number can be used.a
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The Integral of 1/xp
1
1 is convergent if 1 and divergent if 1
pdx p p
x
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Type 2: Discontinuous Integrals
Definition of an Improper Integral of Type 2
If f is continuous on [a,b) and is discontinuous at b, then
if this limit exists (as a finite number)
( ) lim ( )b t
t ba a
f x dx f x dx
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Type 2: Discontinuous Integrals
b) If f is continuous on (a,b] and is discontinuous at a, then
if this limit exists (as a finite number)
( ) lim ( )b b
t aa t
f x dx f x dx
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Type 2: Discontinuous Integrals
The improper integral is called convergent if the corresponding limit exists and divergent if the limit does not exist.
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Type 2: Discontinuous Integrals
c) If f has a discontinuity at c, where a < c <b, and both are convergent, then we define
( ) and ( )c b
a c
f x dx f x dx
( ) ( ) ( )b c b
a a c
f x dx f x dx f x dx
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A Comparison Test for Improper Integrals
Suppose that f and g are continuous functions with ( ) ( ) 0 for f x g x x a
a) If ( ) is convergent, then ( ) is convergenta a
f x dx g x
b) If ( ) is divergent, then ( ) is divergenta a
g x dx f x