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Test the improper integral ∫ log x / x1/2 for convergence

Test the following improper integral for convergence:

    \[ \int_{0^+}^1 \frac{\log x}{\sqrt{x}} \, dx. \]


The integral converges.

Proof. We can compute the value of the improper integral directly. First, we can evaluate the indefinite integral using integration by parts with

    \begin{align*}  u &= \log x & du &= \frac{1}{x} \\ dv &= x^{-\frac{1}{2}} & v &= 2 \sqrt{x}. \end{align*}

Therefore,

    \begin{align*}  \int \frac{\log x}{\sqrt{x}} \, dx &= uv - \int v \, du \\[9pt]  &= 2 \sqrt{x} \log x - \int \frac{2\sqrt{x}}{x} \, dx \\[9pt]  &= 2 \sqrt{x} \log x - 2 \int x^{-\frac{1}{2}} \, dx \\[9pt]  &= 2 \sqrt{x} \log x - 4 \sqrt{x}. \end{align*}

So, to evaluate the improper integral we take the limit,

    \begin{align*}  \int_{0^+}^1 \frac{\log x}{\sqrt{x}} \, dx &= \lim_{a \to 0^+} \int_a^1 \frac{\log x}{\sqrt{x}} \, dx \\[9pt]  &= \lim_{a \to 0^+} \left( 2 \sqrt{x} \log x - 4 \sqrt{x} \right)\Bigr \rvert_a^1 \\[9pt]  &= \lim_{a \to 0^+} \left( -4 - 2 \sqrt{a} \log a + 4 \sqrt{a} \right) \\[9pt]  &= -4. \qquad \blacksquare \end{align*}

2 comments

    • aaron shreeve says:

      I believe you can rewrite

      *** QuickLaTeX cannot compile formula:
       \sqrt{a}log(a) 
      
      *** Error message:
      Cannot connect to QuickLaTeX server: cURL error 60: Peer's Certificate issuer is not recognized.
      Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
      These links might help in finding solution:
      http://wordpress.org/extend/plugins/core-control/
      http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

      can be rewritten as

      *** QuickLaTeX cannot compile formula:
       \frac{alog(a)}{\sqrt{a}} 
      
      *** Error message:
      Cannot connect to QuickLaTeX server: cURL error 60: Peer's Certificate issuer is not recognized.
      Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.)
      These links might help in finding solution:
      http://wordpress.org/extend/plugins/core-control/
      http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37

      then we can just spam l’Hôpital and find that the limit is, in fact, equal to 0.

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