# MATH 2B Study Guide - Midterm Guide: Partial Fraction Decomposition, Indeterminate FormPremium

by OC537488

Department

MathematicsCourse Code

MATH 2BProfessor

AllStudy Guide

MidtermThis

**preview**shows page 1. to view the full**5 pages of the document.**Section 7.5, 7.8 - Improper Integrals

Reviewing the definition of an integral

●

() =

()

●What does this mean?

○The integral of the function ()on the interval from to ,

○Is the limit of the sum of () multiplied by the change of x

●Whenever we have some number in [,] that is NOT in the domain of (), the

integral is said to be improper

Example #1

●

○We know this is improper because of the boundary of

○How do we solve this?

○Think about replacing with a different variable

○

■ We can take the limit as T goes to

■ And then replace with T in the integral to make it possible to solve for

■ After taking the integral, we would then take the limit

○

○

( 1 )

○

(

)

■ Then we can take the limit after simplifying

○

1

■ The limit as 1 approaches is just 1

■ The limit as

approaches is 0

○1 0 = 1

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○So the answer is 1

■ Because the limit is a definitive number, and not an answer such as +,

or , it is said to convergent

Example #2

●

○

○

||( 1 )

○

|| |1|

○

|| 0

■ The limit as T approaches on || is +

■ Because the answer is NOT a definitive number, the limit is said to be

divergent

Example #3

●

○

■ From here, we notice that we have a not so easy integral to evaluate

■ Use integration by parts

○ = , = (1)

○ = , =

○

( 0)

○

( 0) ( 0)

○

(0) ()

○

■ This is an indeterminate form

■ So to solve this problem , we will have to use L’Hopital’s Rule

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