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Improper Integrals - Lecture Notes - Calculus II | MATH 1920, Study notes of Calculus

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 5.10
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)
If fis continuous for xa, then a
fxdx lim
ta
tfxdx (provided the limit exists)
If fis continuous for xb, then −
bfxdx lim
t− t
bfxdx (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.
If fis continuous for all real numbers, then −
fxdx −
afxdx a
fxdx provided both the
integrals on the left converge.
Examples:
1.1
1
x2dx
2.1
1
xdx
3.1
1
xpdx
4.−
0exdx
5.1
1xexdx
6.−
ex
1e2xdx
-3 -2 -1 1 2 3
0.2
0.4
0.6
x
y
Graph of yex
1e2x
pf2

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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 )

 If f is continuous for x  a, then a

fx dx  lim t→ a

t fx dx (provided the limit exists)

 If f is continuous for x  b, then  −

b

fx dx  tlim→− t

b fx 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.

 If f is continuous for all real numbers, then 

−

fx dx  

−

a

fx dx  

a

 fx dx provided both the integrals on the left converge.

Examples:

x^2

dx

x dx

x p^ dx

0 e x^ dx

1

  1 − xe−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

INTEGRALS WITH VERTICAL ASYMPTOTES ( TYPE 2 )

 If f is continuous on a, b and has an infinite discontinuity at b, then a

b fx dx  lim t→b−^

a

t fx dx

 If f is continuous on a, b and has an infinite discontinuity at a, then a

b fx dx  lim t→a^

t

b fx 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.

 If f is continuous on a, b, except for at some c in a, b where f has an infinite discontinuity, then

a

b

fx dx  

a

c

fx dx  

c

b fx dx

provided these last two integrals are convergent.

Examples:

(^1) dx (^3) x

0

(^2) dx x^3

3. ^ − 1

(^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 fx ≥ gx ≥ 0 for x ≥ a.

1. a. If a

fx dx converges then a

 gx dx converges.

b. If a

gx dx diverges then a

 fx dx diverges.

Example:

 (^1)  e x x dx^ is divergent since

1  e x x ^

x for^ x^ ≥^ 1 and^  1

x dx^ is divergent.

1

x^2  5

dx is convergent since 1 x^2

x^2  5

for x ≥ 1 and 

1

x^2

dx is convergent.