https://math.codidact.com/categories/41/tags/5521.rssNew Posts Tagged 'improper-integral' - MathematicsMathematics - Codidact2024-01-07T16:05:42Zhttps://math.codidact.com/posts/290508Why is $ \int_0^{\infty}\left|\frac{\sin x}{x}\right|\ dx=\infty$?Snoopyhttps://math.codidact.com/users/568172024-01-07T15:51:57Z2024-01-07T16:05:42Z<blockquote>
<p><strong>Question</strong>: Why is
$$
\int_0^{\infty}\left|\frac{\sin x}{x}\right|\ dx=\infty\quad ?
$$</p>
</blockquote>
<p>There are several other ways to state the fact in t...https://math.codidact.com/posts/287492Example of $f:[0,1]\to\mathbf{R}$ with $\lim_{a\to 0^+}\int_a^1f(x)dx=L $ for some real number $L$ but $\int_0^1|f(x)|dx=\infty $Snoopyhttps://math.codidact.com/users/568172022-11-27T19:08:16Z2022-11-27T23:35:05Z<p>In the Wikipedia article on <a href="https://en.wikipedia.org/wiki/Improper_integral#Types_of_integrals">improper integrals</a>, the function $f(x)=\frac{\sin x}{x}$ gives an example that is imp...https://math.codidact.com/posts/283082differentiate under integral sign says something went wrongdeleted user#2021-08-03T09:44:06Z2021-08-03T13:08:00Z<p>$$I=\int_0^\infty \frac{\tan^{-1} ax \tan^{-1}bx}{x^2}\mathrm dx$$
$$\frac{dI}{da}=\int_0^{\infty} \frac{\tan^{-1} bx}{x^2} \cdot \frac{x}{(ax)^2+1}\mathrm dx$$
$$\frac{d^2I}{\mathrm da \mathr...