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Homework Help: Improper Integrals - converge or diverge

  1. Mar 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Hello. I have some questions on this assignment, I'm wondering if I could get some help:
    Determine whether the integral converges and, if so, evaluate the integral.

    1) [tex] \int_{e}^{\infty} \frac{dx}{xlnx} [/tex]



    2) [tex] \int_{1}^{4} \frac{dx}{x^2 - 4} [/tex]



    3) [tex]\int_{e}^{\infty} \frac{dx}{(lnx)^2}[/tex]



    4) [tex]\int_{2}^{\infty} \frac {dx}{x^2 + sinx} [/tex]




    2. Relevant equations



    3. The attempt at a solution

    1) I integrate and get [tex] \lim_{b \to \infty} \int_{e}^{b} \frac{dx}{xlnx} = \lim_{ b \to \infty} [ \frac{ln(xlnx)}{lnx}]_{e}^{b} [/tex] ?

    2) It has discontinuity at x = 2 and x = -2 so I evaluate [tex] \int_{1}^{2} \frac{dx}{x^2 - 4} \int_{2}^{4} \frac{dx}{x^2 - 4} [/tex]

    I integrate and get [tex] \frac{-1}{4} \lim_{c \to \infty} \int_{1}^{c} \frac{1}{x+2} - \frac{1}{x-2} dx [/tex]

    I sub in and get ln0?

    3) Does this integrate into [tex] \lim_{b \to \infty} [\frac {-x}{lnx}]_{e}^{b} [/tex] ?

    4) I don't know how to integrate this
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 17, 2010 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Your antiderivative is incorrect. Try u = ln(x)

    Don't you need a + sign between them?

    Where did [itex]c \rightarrow \infty[/itex] come from? Your intgerals are improper at x = 2.
    No.

    To integrate 1/(x-a) try a u substitution.
     
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