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Math Help - Improper Integral

  1. #1
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    Improper Integral

    Consider the integral: ln[x]/x^2 from 1 to infinity. Explain why it does not converge or evaluate the integral.

    Using integration by parts, I eventually got [1/3]x^3 ln[x]-[1/9]x^3 from 1 to t as t approaches infinity. I know the answer is one, but using this I'll get infinity-infinity-0+1/9 when I substitute back in. Could someone help me out?
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  2. #2
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    \int \frac{\ln{x}}{x^2} \, dx<br />

    u = \ln{x}

    dv = \frac{1}{x^2} \, dx

    du = \frac{1}{x} \, dx

    v = -\frac{1}{x}

    \int \frac{\ln{x}}{x^2} \, dx = -\frac{\ln{x}}{x} + \int \frac{1}{x^2} \, dx


    \lim_{b \to \infty} \int_1^b \frac{\ln{x}}{x^2} \, dx =

    \lim_{b \to \infty} \left[-\frac{\ln{x}}{x} - \frac{1}{x}\right]_1^b =

     \lim_{b \to \infty} \left(-\frac{\ln{b}}{b} - \frac{1}{b}\right) - \left(-1\right) = 1
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