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Math Help - Improper Integrals!!!!!!

  1. #1
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    Improper Integrals!!!!!!

    Hi! The problem is: determine whether or not the integral is convergent or divergent, and evaluate if convergent: The integral from negative infinity to negative one of e^-4x dx. I thought that the answer might be zero, because the graph approaches zero, but that might be positive infinity. Please help, I'm really not getting improper integrals!! And also, how do you know if an integral is improper?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by elocin View Post
    Hi! The problem is: determine whether or not the integral is convergent or divergent, and evaluate if convergent: The integral from negative infinity to negative one of e^-4x dx. I thought that the answer might be zero, because the graph approaches zero, but that might be positive infinity. Please help, I'm really not getting improper integrals!! And also, how do you know if an integral is improper?
    An integral is improper when:

    1) One/Both of the limits contain \infty. Ex: \int_{-\infty}^{\infty}e^{-x^2}\,dx


    2) When one of the limits exists at a undefined value of a function. Ex: \int_0^5\frac{1}{x}\,dx

    3) When the integral is being over an interval in which an asymptotic discontinuity occurs. Ex: \int_{-7}^{5}\frac{4}{x-3}\,dx

    Now, to your question:

    \int_{-\infty}^{-1}e^{-4x}\,dx=\left.\left[-\frac{1}{4}e^{-4x}\right]\right|_{-\infty}^{-1}=-\frac{1}{4}e^4+\frac{1}{4}e^{\infty}=\infty

    Thus, the integral is divergent.

    Hope this makes sense!

    --Chris
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  3. #3
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    Thank you! That does make a lot of sense!!
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