Why Do Some Improper Integrals Diverge?

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SUMMARY

This discussion focuses on the convergence of improper integrals, specifically evaluating the integrals of 1/(x-2)^5 from -∞ to 1 and e^x from 0 to ∞. The first integral diverges due to a discontinuity at x=2, while the second integral diverges as well, despite initial calculations suggesting convergence. The confusion arises from improper handling of limits and the conditions under which integrals are split. Proper understanding of discontinuities and limits is essential for accurate evaluation.

PREREQUISITES
  • Understanding of improper integrals
  • Knowledge of limits in calculus
  • Familiarity with convergence and divergence criteria
  • Basic integration techniques
NEXT STEPS
  • Study the properties of improper integrals and their convergence criteria
  • Learn about discontinuities and their impact on integration
  • Review limit evaluation techniques in calculus
  • Practice solving various improper integrals with different discontinuities
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Students preparing for calculus exams, particularly those focusing on improper integrals and convergence, as well as educators teaching advanced calculus concepts.

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Homework Statement


Well I am studying for my final which is in a couple of days, and I am stuck on this topic of convergence of improper integrals. I've been doing a couple and I thought I was doing them correctly but my answers do not go with the answers I was given. So I am stressing out over it. Here are the questions I have:

Determine whether the following integrals converge or diverge, evaluate those that converge.

1) Integral from -inf to 1 of: dx/(x-2)^5

2) Integral from 0 to inf of: e^x dx


Homework Equations


Just the standard limit and integral equations


The Attempt at a Solution


1) I split the integral into two pieces, a) the int from -inf to 0: 1/(x-2)^5 and b) the int from 0 to 1: 1/(x-2)^5 I then took the limit of each integral, so I got:
a) lim as t approaches -inf of a. solving the limit I got (1/-2 - 1/inf) which = -1/2 and is convergant.

b) lim as t approaches 0 of b. solving that limit I came up with a divergent answer. making the original Integral divergent. however the answer I was given says that the integral is convergent. I am confused on this.
-------------------------------------------------------------------------
2) integral from 0 to inf of e^x dx
I split this integral into two pieces as well. a) integral from 0 to 1: e^x dx and b) integral from 1 to inf: e^x dx

Solving each part, I got two convergent answers, but the answer said that its divergent. I am wrong again .

Please help me out, and explain why. I have the answers, that's not what I am going for..I just want to understand this so I can do well on my final. Thanks guys
 
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I didn't get why you are splitting things.
You only do that when you have a discontinuity like going from -1 to 1 for 1/x.

1/(x-2)^2 .. gives -1/(x-2) ] from t to 1, and take limit of t as it appraches -inf

same thing for e^x
 
b) the int from 0 to 1: 1/(x-2)^5 I then took the limit of each integral, so I got: ...

also, you did this step wrong!
you don't have any inf/discountinity from 0 to 1
 

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