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Math Help - Improper Integrals

  1. #1
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    Improper Integrals

    I'm having trouble starting the following problem...

    Explain why the integral is improper.

     \int_1^4 \frac{dx}{x^2 lnx}
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Because ln(1)=0 making the integrand have a discontinuity at x=1
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    Quote Originally Posted by Mathstud28 View Post
    Because ln(1)=0 making the integrand have a discontinuity at x=1
    Wouldn't that just mean that you subtract that 0 from the function when 4 is plugged in? This may sound a little confusing.
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    and If I am not mistaken

    all you do is take the integral except make the limits of integration lim[b as b approaches 1] and 4 and if it still doesnt work it diverges
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Haha it does

    but...hmmm If I think what you are saying is what you are really saying then the answer is no...you cant just throw out an aspect in lieu of an easier integral
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  6. #6
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    Quote Originally Posted by larson View Post
    Wouldn't that just mean that you subtract that 0 from the function when 4 is plugged in? This may sound a little confusing.
    You need to read the first few lines of this before you'll understand the answer to your question.
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    Ok so another easy way to tell if it is improper is to just graph the function and see if you get any horizontal lines (aka errors) within the graph.
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    Quote Originally Posted by larson View Post
    Ok so another easy way to tell if it is improper is to just graph the function and see if you get any horizontal lines (aka errors) within the graph.
    No. Improper if the integrand has vertical asymptotes in the interval of integration.
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  9. #9
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    Quote Originally Posted by mr fantastic View Post
    No. Improper if the integrand has vertical asymptotes in the interval of integration.
    Errr... thats what I meant :\ sorry. Thanks though for the correction.
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  10. #10
    MHF Contributor Mathstud28's Avatar
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    IT is like I said

    if the integrand (The thing you are taking the integral of) has a discontinuity that is an element of [a,b] where a and be are the limits of integration then the integral is improper
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  11. #11
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    Quote Originally Posted by Mathstud28 View Post
    if the integrand (The thing you are taking the integral of) has a discontinuity that is an element of [a,b] where a and be are the limits of integration then the integral is improper
    No. Consider f(x) = \left\{ \begin{array}{c}-1 \mbox{ for }-1\leq x\leq 0 \\ 1\mbox{ for }0<x\leq 1 \end{array}\right. then \int_{-1}^1 f(x) dx is not an improper integral.
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  12. #12
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    Quote Originally Posted by Mathstud28 View Post
    if the integrand (The thing you are taking the integral of) has a discontinuity that is an element of [a,b] where a and be are the limits of integration then the integral is improper
    What TPH is true. Note that not all discontinuities are vertical asymptotes ......
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