Improper Integral
Last updated
Last updated
After learning Definite Integral, Indefinite Integral, now it's Improper Integral. The major difference between them is their Boundaries.
The
improper integralmeans 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 IntegralRefer to video from ProfRobBob: Improper Integrals 5 Examples
There're 6 cases of different improper integral:
Case 1: From a constant to positive infinity.
Case 2: From negative infinity to a constant.
Case 3: From negative infinity to positive infinity.
Case 4: From 0 to e.
Case 5: From a constant to a constant, but has an infinite discontinuity.
Case 6:
Convergent & DivergentWe 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 IntegralsBasic 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
Rewrite the improper integral to limit form:
Do the basic calculation.
The key point is:
Solve:
Solve:
Solve:
