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Improper Integrals

  1. Oct 23, 2011 #1
    Just got into improper integrals, in my Calculus 2 class. We're looking to see if the integral converges or diverges.
    1. The problem statement, all variables and given/known data
    The integral given:
    ∫(dt/(t+1)^2) on the interval from -1 to 5


    2. Relevant equations
    uhhh...



    3. The attempt at a solution
    Took the limit as "a" goes to -1.

    Did a simple u substitution with u=t+1, so that du=dt.

    So, you're left with (du/u^2)

    The integral of that is -(1/u) meaning -(1/(t+1))

    Then I used the fundamental theorem of calculus by evaluating the integral from 5 to "a".

    That looks like: -(1/(5+1)) - (1/(a+1))

    Basically, I have no clue if I'm doing this right. According to the back of the book, it diverges. But I have no idea how to see that. Any help would be appreciated. Thanks.
     
  2. jcsd
  3. Oct 23, 2011 #2

    dynamicsolo

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    Homework Helper

    You have
    [tex]\int_{-1}^{5} \frac{dt}{(t+1)^{2}} = -\frac{1}{6} - [ \lim_{a \rightarrow -1} (-\frac{1}{a+1}) ] . [/tex]

    What happens when you apply the limit?
     
  4. Oct 23, 2011 #3
    You mean apply the -1 into "a"? That would make it undefined because you'd be dividing by zero. But what does this tell me about convergence/divergence?
     
  5. Oct 23, 2011 #4

    dynamicsolo

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    Homework Helper

    And so that limit is undefined. Therefore, there is no meaningful value for the integral: that is what is meant by "divergence". We say the integral "converges" if the limit for its value approaches a finite number. If the limit for the integral does not approach a finite value, or does not even exist, the integral is said to "diverge". (It should also give the definitions of convergence and divergence of an integral in your textbook.)
     
  6. Oct 23, 2011 #5
    Wow. Thanks for clearing that up. I'm starting to understand it now, as I do more problems. Sometimes, I can't understand my textbook, nor my professor.
     
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