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Integrate improper integral with infinite discontenuities

  1. Oct 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Integrate the improper integral (use correct notation). State whether it's converging or diverging.
    10
    ∫ 7/(x-9)^2 dx
    8
    2. Relevant equations

    b c
    ∫ f(x) dx= lim ∫ f(x) dx
    a c → d a


    3. The attempt at a solution

    b
    lim ∫ 7(x-9)^2 dx
    c → 10- 8

    Let u= x-9
    -du= dx

    lim ∫ -u^2
    c→10-

    lim -2u^2
    c→10-

    lim -2(x-9)^2
    c→10-

    lim -2(10-9)^2-2(8-9)^2
    c→10-

    I know I must be doing something wrong because the answer is ∞, any ideas where I went wrong?
     
  2. jcsd
  3. Oct 30, 2012 #2

    Mark44

    Staff: Mentor

    Mod note: Added [ code ] tags to preserve the OP's spacing.
    The problem is not at 10 - it's at 9, which is where the denominator becomes zero. You're going to have to split the interval [8, 10] into two parts, and then deal with each separately.
     
  4. Oct 30, 2012 #3
    Ok, so here's what I've got.
    9 10
    ∫ 7/(x-9)^2 dx + ∫ 7/(x-9)^2
    8 9

    9 9
    ∫ 7/(x-9)^2 dx = lim ∫ 7/(x-9)^2 dx
    8 x→9 8


    10 9
    ∫ 7/(x-9)^2 dx = lim ∫ 7/(x-9)^2 dx
    8 x→9 8

    10
    ∫ 7/(x-9)^2 dx
    8

    Am I on the right track?
     
    Last edited: Oct 30, 2012
  5. Oct 30, 2012 #4

    Mark44

    Staff: Mentor

    The above is correct.
    The limit should be taken as b →9-, not as x →9. The upper limit of integration should be b.

    What you have just below is just the repeat of the line above.
     
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