Test the following improper integral for convergence:
![\[ \int_{0^+}^1 \frac{\log x}{\sqrt{x}} \, dx. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-67e495fd6a3b5173214e651fffa09464_l3.png "Rendered by QuickLaTeX.com")
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
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*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-41b6b2b1f2f6963d2886529d9813e602_l3.png "Rendered by QuickLaTeX.com")
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*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-010fd24ca424ba14201fe204828bad39_l3.png "Rendered by QuickLaTeX.com")
The substitution for
will not give the limit 0, since logarithm is not defined there – it is an indeterminate form 
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=37can 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=37then we can just spam l’Hôpital and find that the limit is, in fact, equal to 0.