Improper Integral
After learning Definite Integral
, Indefinite Integral
, now it's Improper Integral
. The major difference between them is their Boundaries
.
The
improper integral
means the integral's boundary or boundaries are infinite, ∞ (or -∞).
Refer to Khan academy: Introduction to improper integrals Refer to Improper Integrals (KristaKingMath)
It looks so fearful yet not too hard to understand.
Types of Improper Integral
Types of Improper Integral
Refer to video from ProfRobBob: Improper Integrals 5 Examples
There're 6 cases of different improper integral:
Case 1: From a
constant
topositive infinity
.Case 2: From
negative infinity
to aconstant
.Case 3: From
negative infinity
topositive infinity
.Case 4: From
0
toe
.Case 5: From a
constant
to aconstant
, but has aninfinite discontinuity
.Case 6:
Convergent
& Divergent
Convergent
& Divergent
We can call an improper integral
:
Divergent
: When the limit of the improper integral DOES NOT EXIST.Convergent
: When the limit of the improper integral EXISTS.
Solve Improper Integrals
Solve Improper Integrals
Basic Strategy:
Replace the
infinite
as a variable, etc.t
Rewrite the expression as taking the limit of the Integral, whereas the
t → ∞
Calculate the Integral with a normal variable first, and gets the result function.
Calculate the limit of the function
Type 1
Type 2
Rewrite the improper integral to limit form:
Do the basic calculation.
The key point is:
Type 5
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