I have to show the following equation:  $\int_{0}^{\pi} t \cdot f(sin \; t) \; dt = \frac{\pi}{2} \int_{0}^{\pi} f(sin \; t) \; dt$  with $f : [0, 1] \rightarrow \mathbb{R}$ is continuous.  I transformed both sites to the following:  $\int_{0}^{\pi} t \cdot f(sin \; t) \; dt = \pi \cdot F(sin \; \pi)$ and  $\frac{\pi}{2} \int_{0}^{\pi} f(sin \; t) \; dt = \frac{\pi}{2} F(sin \; \pi) - \frac{\pi}{2} F(0)$  Now I don't know how to go on, because I do know nothing about $f(x)$ or $F(x)$. Is my approach correct or is there a other possibility to show the equation?
Let $f(\sin x)$ be a given function of $\sin x$.   How would I show that $\int_0^\pi xf(\sin x)dx=\frac{1}{2}\pi\int_0^\pi f(\sin x)dx$?
Scala Lists of lists of integers How do I resize an encrypted LVM to install another copy of Ubuntu? I have an old PowerBook which has been happily running Ubuntu Server for a few months. Now I want to repartition the hard drive and install Ubuntu Client, while keeping my existing installation intact.  When I first installed Ubuntu, I chose to erase the entire hard drive and use it as an encrypted LVM. Now, the installer seems to recognize the partitions on the drive, but it never prompts me for the password to the LVM.  How do I make room for another installation alongside my current one? How to set the monitor to its native resolution which is not listed in the resolutions list? After installing Ubuntu 10.04 with my Samsung SyncMaster B2030, native resolution (1600X900) is not found in the list of resolutions. I'm writing a thesis which is an aggregate of separate articles. I have the source code for the articles and I'm using \include to include them in the main document of the thesis. So in my main.tex (thesis document), I have  \include{article1} some pages in between the two articles \include{article2}   The problem that I'm getting is with the Appendix sections. In article 1, I have at the end:  \appendix \numberwithin{equation}{section} \section{Coefficients}\label{sec:Coeffic} \subsection{Good coefficients} then comes the first line of appendix   This gives in article 1 (when  compiled separately with the article class):     A. Coefficients      A.1. Good coefficients   and equation numbers in the Appendix A as (A.1), (A.2)  which is ok and what is expected.  When I put article 1 within main.tex, I use \include command. Also, I removed the document class for article 1 and put \chapter instead.  But when I compile main.tex (the thesis document), I get the appendix sections as:     .1 Coefficients      .1.1 Good coefficients   and the equation numbers in the appendix A as (.1.1), (.1.2) etc  My article 1 is treated as a chapter in the main thesis document (main.tex) and the title and section numbers, equation numbers are preceded by A in main.tex. This is what I want.  Maybe it would be ok to get the appendix in my main.tex as     A.A. Coefficients      A.A.1 Good coefficients   and the equation numbers in the Appendix as (A.A.1), (A.A.2) etc  to denote that it refers to appendix A of paper A. First A is for the first article and second A is for first appendix. How to get this? Firstly, I apologise if my thinking is completely incorrect - I am by no means a physicist (yet!), I have included layman illustrations to hopefully assist in explaining my thinking.  I am imagining the big bang to be, for lack of a better word, an explosion, thus the universe to be progressively expanding:    Therefore, to me it seems logical to suggest that there is an edge.  If this is the case, then is it reasonable to suggest that explosion + edge = 'bright edge'? Where light projects in all directions at the edge, not just outwards into the 'nothingness'.     So, this brings me to my question: why, when someone looks deeply enough 'at the edge' do they see an absence of light?   Again, apologies if I'm jumping to irrational conclusions &amp; am way off. I was curious about how other people use the this keyword. I tend to use it in constructors, but I may also use it throughout the class in other methods. Some examples:  In a constructor:  public Light(Vector v) {     this.dir = new Vector(v); }   Elsewhere  public void SomeMethod() {     Vector vec = new Vector();     double d = (vec * vec) - (this.radius * this.radius); } Is there a way to step between 0 and 1 by 0.1?   I thought I could do it like the following, but it failed:  for i in range(0, 1, 0.1):     print i   Instead, it says that the step argument cannot be zero, which I did not expect. I worked on a Word document and also saved the file on my flash drive.  When it asked me to save, I pressed the Replace document option by mistake.    I can see the missing work under the Properties menu. Will I be able to recover the lost work? Possible Duplicate:          The  describes the expansion of the $n$th power of the sum $(a+b)$:  $$(a+b)^n = \sum_{k = 0}^n {n\choose k}a^kb^{n-k}$$  In calculus, there is a generalization of the product rule called , which describes the expansion of the $n$th derivative of the product of two functions $f$ and $g$:  $$(fg)^{(n)} = \sum_{k = 0}^n {n\choose k}f^{(k)}g^{(n-k)}$$  It's not difficult to prove using the product rule and induction, what I'm really trying understand here is why the formula is true. Is there a way to prove it using the binomial formula? How are these two formulas related? Is there a combinatorial proof for Leibniz's rule like there is for the ?   I understand this question is a bit vague, but I hope you'll agree that these two formulas look so similar it just feels like there should be some relation between the two. Of course this is not necessarily true, but I'm hoping it's not just a crazy coincidence. Running own executable on Android shell I need a specific command line tool and I have made a C program in my Linux shell. I have compiled the program with an ARM cross-compiler. I have then moved the program into the Android file system and tried to run it.  The output is permission denied.  What do I have to do, in order to run my own compiled programs in Android file system? How do I center text horizontally and vertically in a TextView? No secondary button (right click) on touchpad I'm experiencing a serious problem with the new Ubuntu 18.04:  System Dell XPS 13 with a freshly installed Ubuntu 18.04 LTS.  Problem description The touchpad provides only primary button (single click and double click), but no secondary button. In particular, when using the "test Your Settings" option in Settings -> Devices -> Mouse &amp; Touchpad, any click on the touchpad is signaled as primary button, no matter where I click (left or right side of the touchpad). This remains the case even after I change the Primary Button from Left to Right (in Settings -> Devices -> Mouse &amp; Touchpad -> General).  In comparison, an external mouse connected to my laptop (via usb) works fine: it provides both primary and secondary buttons with sides corresponding to the choice of Primary Button (in Settings -> Devices -> Mouse &amp; Touchpad -> General).  This may be related to   Any ideas?  Update: Meanwhile, I found out that in addition to the solution proposed by EntropyWins, one can also control this touchpad behavior via the Tweaks tool (a.k.a. gnome-tweaks) under Keyboard &amp; Mouse -> Touchpad -> Mouse Click Emulation. In any case, as EntropyWins explains in his answer, this is not a bug. Wine or virtual machine (VirtualBox) to run apps (eg. Photoshop) I am wondering if it will be better (stability, convenience and speed) to run apps like Photoshop with Wine or a virtual machine like VirtualBox? Prove $ \sum \frac{\cos n} { \sqrt n}$ converges How can I increment a date by one day in Java? Sounds simple but I seem to fail every time, that is it turns into scrambled eggs instead. How do you make it so that you can flip it over both without it breaking and before it burns? Origins of possessive pronouns I am using a plugin to help "build" a customized product. There are different tabs/steps the user has to go through with back and continue buttons. At the last step I would like to add an "agree to terms" box with a checkbox "Agree". I have this done. But I would like to hide the add to cart button in Woocommerce until the user selects the checkbox agreeing to the terms.  This will be beneficial to me to make sure they reviewed the return policy as well as avoid confusion with the add to cart button appearing during all the steps, assuring they go through all the steps.  Here is the checkbox coding i pulled from console:  &lt;ul data-rules="{&amp;quot;Agree_0&amp;quot;:[&amp;quot;&amp;quot;]}" data-original-rules="{&amp;quot;Agree_0&amp;quot;:[&amp;quot;&amp;quot;]}" data-rulestype="{&amp;quot;Agree_0&amp;quot;:[&amp;quot;&amp;quot;]}" data-tm-validation="{&amp;quot;required&amp;quot;:true}" class="tmcp-ul-wrap tmcp-elements tm-extra-product-options-checkbox tm-element-ul-checkbox element_16 termsbox-ul"&gt;     &lt;li class="tmcp-field-wrap tmhexcolorimage-li-nowh"&gt;         &lt;label for="tmcp_choice_16_0_35"&gt;    &lt;input class="tmcp-field termsbox tmhexcolor_16_0_35 tm-epo-field tmcp-checkbox" name="tmcp_checkbox_16_0" data-limit="" data-exactlimit="" data-minimumlimit="" data-image="" data-imagec="" data-imagep="" data-imagel="" data-image-variations="[]" data-price="" data-rules="[&amp;quot;&amp;quot;]" data-original-rules="[&amp;quot;&amp;quot;]" data-rulestype="[&amp;quot;&amp;quot;]" value="Agree_0" id="tmcp_choice_16_0_35" tabindex="35" type="checkbox"&gt;     &lt;span for="tmcp_choice_16_0_35"&gt;&lt;/span&gt;&lt;span class="tc-label tm-label"&gt;Agree&lt;/span&gt;&lt;/label&gt; &lt;span class="price tc-price  hidden"&gt;     &lt;span class="amount"&gt;$0.00&lt;/span&gt; &lt;/span&gt;&lt;i data-tm-tooltip-html="This is were the description will be to agree" class="tm-tooltip tc-tooltip tcfa tcfa-question-circle"&gt;&lt;/i&gt;     &lt;/li&gt;&lt;/ul&gt;   Here is the add to cart section with the quantity  &lt;div class="quantity buttons_added"&gt;     &lt;input value="-" class="minus button is-form" type="button"&gt;    &lt;input class="input-text qty text" step="1" min="1" max="9999" name="quantity" value="1" title="Qty" size="4" pattern="[0-9]*" inputmode="numeric" type="number"&gt;     &lt;input value="+" class="plus button is-form" type="button"&gt;  &lt;/div&gt; &lt;button type="submit" name="add-to-cart" value="267" class="single_add_to_cart_button button alt"&gt;Add to cart&lt;/button&gt;   I am self taught with html and css, but javascript is to foreign for me  I tried the following in CSS but it does nothing.Tried is on codepen and works (i know needs to show instead of hide when checked)....  input[type=checkbox]:checked + .single_add_to_cart_button {     display:none !important;  } Pacf lag axe , is not an integer Boxplot interpretation: is it correct that a boxplot is missing a whisker? I have a data set that has customerID and daily visits. This dataset records the number of times customers visited the site in a day.   customerID   TotalVisits  1            6  2            17  3            20  4            25   Just to explore the data further, I tried plotting the TotalVisits values in a box plot. The plot doesn't seem to have a lower whisker. q1=1 and there are no values between q1 and  q1-1.5(IQR). The plot has q1, q2 and q3. The minimum value and q1 are the same.   Is this why the boxplot doesn't have the lower whisker? Could you please help me with the interpretation? Is it correct to draw a boxplot without any whiskers?  [I am new to the site and I am not able to post the image.]