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Math Help - Show that this integral is divergent

  1. #1
    s3a
    s3a is offline
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    Show that this integral is divergent

    The question and my work are both attached.

    Can someone please help me understand what's going on?

    Any help would be greatly appreciated!
    Thanks in advance!
    Attached Thumbnails Attached Thumbnails Show that this integral is divergent-temp_q.jpg   Show that this integral is divergent-mywork.jpg  
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  2. #2
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    You correctly applied the definition of the improper integral by splitting up the integral into two parts:

    \int_{-\infty}^{\infty}x\ dx=\lim_{t \to \infty}\int_{-t}^{0}x\ dx+\lim_{t \to \infty}\int_{0}^{t}x\ dx

    But you need to evaluate the two limits independently:

    \lim_{t \to \infty}\int_{-t}^{0}x\ dx=\lim_{t \to \infty}(\frac{1}{2}0^2-\frac{1}{2}(-t)^2)=\lim_{t \to \infty}-\frac{1}{2}t^2=-\infty

    and

    \lim_{t \to \infty}\int_{0}^{t}x\ dx=\lim_{t \to \infty}(\frac{1}{2}t^2-\frac{1}{2}(0^2))=\lim_{t \to \infty}\frac{1}{2}t^2=\infty

    You combined the two limits after integrating but before taking the limit, and the t^2 terms canceled out.

    Post again in this thread if you're still having trouble.
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