Improper Integrals - Calculus II - Lecture Slides, Slides of Calculus

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6.6 Improper Integrals

Definition of an Improper Integral of Type 1

a) If exists for every number t ≥ a, then

provided this limit exists (as a finite number).

b) If exists for every number t ≤ b, then

provided this limit exists (as a finite number).

The improper integrals and are called

convergent if the corresponding limit exists and divergent

if the limit does not exist.

c) If both and are convergent, then we define

∫

t a

f ( x ) dx

∫

b t

f ( x ) dx

∫ ∫

∞

→∞

=

t a (^) t a

f ( x ) dx lim f ( x ) dx

∫− (^) ∞ ∫ →−∞

=

b t

b

t

f ( x ) dx lim f ( x ) dx

∫

∞ a ∫− (^) ∞ f^ ( x ) dx

a f ( x ) dx

∫

∞ a

f ( x ) dx ∫ −∞

b f ( x ) dx

∫ ∫ ∫

∞

−∞

∞

− ∞

= + a

a f ( x ) dx f ( x ) dx f ( x ) dx

= = [ ] = [ − ] = ∞ →∞ →∞ →∞

∞ ∫ ∫ ln ln ln^1

1 lim^1 dx lim x^1 lim t

x

dx

x t

t

t

t

t

An example of a divergent integral:

The general rule is the following:

is convergentif p 1 anddivergentif p 1

1

1 ∫ > ≤

∞ dx x

p

∫

∞

1 2

isconvergent

Recall from thepreviousslide that dx

x

Definition of an Improper Integral of Type 2

a) If f is continuous on [a, b) and is discontinuous at b , then

if this limit exists (as a finite number).

a) If f is continuous on (a, b] and is discontinuous at a , then

if this limit exists (as a finite number).

The improper integral is called convergent if the

corresponding limit exists and divergent if the limit does

not exist.

c) If f has a discontinuity at c, where a < c < b, and both

and are convergent, then we define

∫ (^) − ∫ →

=

t a

b a (^) t b

f ( x ) dx lim f ( x ) dx

∫ (^) + ∫ →

=

b t

b a (^) t a

f ( x ) dx lim f ( x ) dx

∫

b c ∫ f^ ( x ) dx

c a

f ( x ) dx

∫

b a

f ( x ) dx

∫ =^ ∫ + ∫

b

c

c

a

b

a

f ( x ) dx f ( x ) dx f ( x ) dx Docsity.com