The integral is as:     $$\int_0^\infty\sin\left(x\right)\times x^{n}\;\mathbb{d}x$$   Where $n\in\mathfrak{R}$.   I was able to find the convergence of this integral using Dirichlet's test, which implied $-2&lt;n&lt;0$.  But the hard part is to find the value of this integral for me. Wolfram alpha gives the answer as:   Which is very close to gamma function. So i tried to assume integral of form $$\int_0^\infty e^{-ix}\times x^n\;\mathbb{d}x$$  So that i would take the $-\mathfrak{Im}$ part of this integral. But I still have no idea how to evaluate this.
I'm looking for several ways to prove that $$\int_{0}^{\infty }\sin(x)x^mdx=\cos\left(\frac{\pi m}{2}\right)\Gamma (m+1)$$  for $-2&lt; Re(m)&lt; 0$
Cursor hitting "border" between differently sized monitors Can I apply for a Schengen visa of the first port of entry or does it need to be the country of the longest duration? I'm planning to travel to the Netherlands, Spain and Bulgaria. My longest stay will probably be in Bulgaria. Now, Bulgaria doesn't issue a Schengen visa but allows people with a Schengen visa to visit. Which visa should I be applying for? Bulgarian Visa AND Schengen visa or just the Schengen visa for Netherlands or Spain? How can I find out the variable name in python? I know how I can find out the python command by clicking a button in the Blender interface, and then look in the info window:   For example, as can be seen, I added a cube, UV sphere, plane, camera, lamp, pressed A twice.  But how can I find out an object variable name, for example the y-rotation of an object. Because I can't find the y-rotation, when I point my mouse over it, I can only find the rotation: I'm using pdflatex, and use \includegraphics to include content of figures from PDF. PDF are exported from InkScape.  \begin{figure} [!h]     \centering     \includegraphics[height=3cm]{fig/SUSAN.pdf} \end{figure}   Included PDF or source SVG has same colors. Green has always RGB value (0, 255, 0). Problem occurs when I include this files from LaTeX. In output PDF on some pages colorspace is darker. Even same image on other page has different colourity.  My PDF  . As you can see on page 7 (11 in PDF) USAN areas are bright green, but then on page 5 (9 in PDF) are green points on corners more darker. HTML code for an apostrophe Seemingly simple, but I cannot find anything relevant on the web.  What is the correct HTML code for an apostrophe? Is it &amp;#8217;? What is the difference between dpkg and aptitude/apt-get? When installing a software via dpkg -i packageA.deb, will the dependencies required by packageA be downloaded and installed automatically? How is this different from using apt-get or aptitude? Play audio with Python Install Headers on Kali apt-get install linux-headers-`uname -r` Reading package lists... Done Building dependency tree Reading state information... Done E: Unable to locate package linux-headers-4.9.0-kali3-amd64 E: Couldn't find any package by glob 'linux-headers-4.9.0-kali3-amd64' E: Couldn't find any package by regex 'linux-headers-4.9.0-kali3-amd64' How to convert standard IP address format string to hex and long? Blender 2.80 system Req + GPU req Can someone please let me know the minimum system requirements and GPU requirement to run blender 2.80 and eevee smoothly?   Also please let me know the optimum system requirements for the purpose of    environment modeling Game development charachter modelling and animation   Also if someone can suggest me a good desktop/laptop for personal use that is compatible with blender 2.80 in a low budget.  Any help will be much appreciated.  Thanks in advance! How to scale a tikzpicture to \textwidth Does umount calls sync to complete any pending writes We have BBB based custom board with kernel 3.12 running on it.  I have doubt regarding umount and &amp; sync.  Lets say a script is umounting a partition, Does it require to run sync command before umount to complete pending writes. ? Number of solutions of an arithmetic function's equation Say, an equation is given below \begin{equation} 2\pi(x) - \pi(2x)=\omega(x) \end{equation}  where $x$ is a positive integer, $\pi(x)$ is the prime-counting function, and $\omega(x)$ is the number of distinct prime factors of $x$.  I would like to know if this equation holds for finitely many values of $x$ or infinitely many.  If one uses asymptotic formula (from number theory) it can be written that \begin{equation} \omega(x) \sim \ln x / \ln \ln(x) \end{equation} (from )  and  \begin{equation} \pi(x) \sim x/\ln(x), \end{equation} so the above equation can be rewritten (after simplification) as \begin{equation} 2x\ln(2)=\ln(x)\ln\ln(x)\ln(2x) +f(x) \end{equation}  where $f(x)$ is an error term.  If  I plot left side (which is linear, supposed to be a straight line) and right side(a curve),assuming $ f(x) $ is 100% accurate, then the equation implies(?!) that the right will intersect left side. Since line can intersect a curve ($ln  $ function) finite times, so there are finite intersection point. So, can I say that there are only finitely many values of $x$?  graph example-   here f(x) is assumed as x + sinx  without the f(x)- What does each AMS package do? In my view, any document with a substantial amount of mathematics should load the packages from AMS-TeX. Hence, most of my document's preambles contain  \usepackage{amsmath, amsthm, amssymb, amsfonts}   (I could add to that the non-AMS mathtools, etc.). What do each of those packages do? Do I really need to list all of them, or do some of them load others? In particular, does any package load all of them, so that I can simply type  \usepackage{ams-everything}   shorter than the current list? What is so special about $\alpha=-1$ in the integral of $x^\alpha$? Of course, it is easy to see, that the integral (or the antiderivative) of $f(x) = 1/x$ is $\log(|x|)$ and of course for $\alpha\neq - 1$ the antiderivative of $f(x) = x^\alpha$ is $x^{\alpha+1}/(\alpha+1)$.  I was wondering if there is an intuitive (probably geometric) explanation why the case $\alpha=-1$ is so different and why the logarithm appears?  Some answers which I thought of but which are not convincing:   Taking the limit $\alpha=-1$ either from above or below lead to diverging functions. Some speciality of the case $\alpha=-1$ are that both asymptotes are non-integrable. However, the antidrivative is a local thing, and hence, shouldn't care about the behavior at infinity. Why is area of a surface of revolution integral $2\pi y~ds$? not '$dx$'? Can we have a list of all the Stack Exchange domains somewhere, for firewall purposes? I work behind an really annoying firewall, and I just want the net admin to add all the SE websites to the whitelist. Right now I've lost access to Stack Overflow's authentication, but not Meta's... so I have to come here before I log in there.  It's driving me nuts. Basically I just want to add an exclusion rule for all SE domains; not only the website domains, but also the content domains such as gravatar.com, quantserve.com, the domain that holds static content, etc... How to copy metadata of Partial Sandbox into a new Developer sandbox? I Have only one license of Partial sandbox. I want to refresh it with Production. Before that I want to copy all the metadata present in the Partial sandbox into New/Existing Developer sandbox. How can do that?  Is it supported by salesforce? or are there any tools available for this requirement.  Thanks. How is a second measured? And why is it measured that way? The earth's rotation around itself (one rotation around Earth's own axis) doesn't take exactly 24 hours. It off by some seconds which becomes somewhere around 6 hours per year and 1 day in 4 years(leap year), which brings the question why didn't we modify the measurement of 1 second ever so slightly so as to avoid leap years altogether.   Well how is 1 second measured exactly?  says      the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. []   Well, why is it measured that way? Is there any technical reason like it is easy to measure and can be standardized easily? Or is it really possible to modify the measurement of time? How to re-install http windows service I replaced apache webserver with http(WINHTTP windows service) for use virtual hosts feature. But I was missing  http(WINHTTP windows service) which I previously removed.I  now want to  reinstall this service for use print spooler.