I need to proof that $$\int_{0}^{\pi}xf(\sin(x))dx=\frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))dx$$ The problem says that the substitution $u=\pi-x$ is useful. Following their advice $u=\pi-x,du=-dx$ $$-\int_{0}^{\pi}(\pi-u)f(\sin(\pi-u))du$$ What do I do next, this the first time I encounter integrals writeen like this and I am not that familiar with proof thanks in advance !
Let $f(\sin x)$ be a given function of $\sin x$.   How would I show that $\int_0^\pi xf(\sin x)dx=\frac{1}{2}\pi\int_0^\pi f(\sin x)dx$?
How do I alter a language item link? I am looking to modify the default output of Drupal 8 of links in a certain situation, the language switcher. Ideally I want to be able to do that in a twig template and not have to produce a  that overwwrites that whole link generation system and add a BIG if for the cases I need to overwrite.  I think the output layout should be controllable in the twig template similar to  (this answer outputs the same url for every languages)  I am using bootstrap as a base theme so i'd like to use the bootstrap markup for my links (btn btn-primary, dropdown)  Here is my code I am trying to make links--language-block.html.twig  {%- if links|length == 2 -%}     {# show only alternate language button #}     {%- for key, item in links -%}         {%- if not item.attributes['#options']['set_active_class'] -%} {# &lt;--- this is always true!? #}             &lt;a href="{{ item.link['#url'] }}" class="btn btn-primary"&gt;{{ item.link['#title'] }}&lt;/a&gt; {# #url is always the same what ever the language it is pointing to! #}         {%- endif -%}     {%- endfor -%} {%- elseif links|length &gt; 2  -%}     {# show selected language in button and other languages in drop down #}     &lt;button class="btn btn-primary dropdown-toggle" type="button" data-toggle="dropdown"&gt;{{ selectedLanguage }}&lt;span class="caret"&gt;&lt;/span&gt;&lt;/button&gt;     &lt;ul class="dropdown-menu"&gt;         {% for key, item in links %}             {%- if not item.attributes['#options']['set_active_class'] -%} {# &lt;--- this is always true!? #}             &lt;li&gt;&lt;a href="{{ item.link['#url'] }}"&gt;{{ item.link['#title'] }}&lt;/a&gt;&lt;/li&gt;             {% endif %}         {% endfor %}     &lt;/ul&gt; {%- endif -%}   Anyone has an idea on how to do this? I have recently created a logo. I have seen an online trend of showing off a logo in a realistic environment, illustrating possible use cases of the logo.  An example of this can be seen here, with the original logo at the bottom and a 3D mockup at the top:    How can I create such a mockup? What does "--" (double-dash) mean? I have seen -- used in the compgen command.  For example:  compgen -W "foo bar baz" -- b   What is the meaning of the -- in there? Activity has leaked window com.android.internal.policy.impl.PhoneWindow$DecorView@44f72ff0 that was originally added here How do you keep user.config settings across different assembly versions in .net? Basically the problem is that each time the assembly version changes (i.e. the user installs a new version of the application) all their settings are reset the the defaults (or more accurately a new user.config file is created in a folder with a different version number as the name)  How can I keep the same settings when upgrading versions, since using ini files or the registry seem to be discouraged?  When we used Clickonce it seemed to be able to handle this, so it seems like it should be able to be done, but I'm not sure how. I am a British citizen and US permanent resident. I rarely need visas. However, I am considering a Mediterranean cruise in the spring or summer of 2019.  Unlike British politicians, I like to plan ahead and be prepared. If I am going to need a Schengen visa, I want to assemble the documentation early next year, so I can apply to the appropriate country as soon as I have an itinerary and it is at most 90 days before travel. Given how things are going, I am preparing for a hard, no-agreement, Brexit.  For trade, that would mean WTO rules. For personal travel, is there a similar fall-back that would allow me visa-free visits to the Schengen countries? Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$ Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$.  Writing out several terms, I think the answer is $1/(n+1)$, but I'm struggling to prove this. I would greatly appreciate it if someone could show how to prove this. I can't get into my field research prize How to move /usr to a new partition? Redshift of merging black holes How did they found that the gravitational waves where emitted at redshift $z=0.09$?  I understand the measurement of redshift for an electromagnetic wave where we have measured in a lab various transitions and therefore we can make a comparison with the wavelength we receive.  But how can they manage to get the redshift for the emitter of gravitational waves, since we have no reference? Inner product on $C(\mathbb R)$ With axiom of choice it is possible to construct an inner product on $C(\mathbb R)$. My question is, is it possible to explicitly construct an inner product on $C(\mathbb R)$? I.e. to give a closed formula to calculate the inner product? I know it is straight-forward to write down a scalar product using a Hamel basis. This is not the answer I am looking for. This question came to me, when a student asked me in the lecture today 'whether there are vector spaces without inner products'. So I tried to find scalar produces for function spaces. I think I managed to write one down for $L^1((0,1))$. But I failed to construct one for $C(\mathbb R)$. Knowing the interface of ubuntu and editing them Where are the files for the desktop or the interface that appears on the screen stored? And how can we edit them. How are they programmed?  What programming language is used to built the interfaces? How can I make a tool drop blocks with CanPlaceOn tag In Adventure mode, I'm trying to let a player get blocks that he can place later, is there a way to do it directly or do I have to change them in his inventory?  Situation: The player gets a tool that will allow him to clear a boulder blocking the road. Further down the path, he encounters a broken bridge, and I want him to use the blocks he gathered before to rebuild it.  At the moment, if I build the boulder with blocks having the CanPlaceOn tag, the tag disappear when the player destroys them and they go in its inventory. Proof of continuity $f(x+y) = f(x) + f(y)$ How can I render an object with an hdr background? I have an object with an hdr for the background. What do I do in Cycles to render the background with my object?  How is this different from an ordinary render in Cycles?  The hdr is a big one, 350MB. How to make bash put prompt on a new line after cat command? What I get:  host:~ user$ cat example.txt some texthost:~ stas$   What I want to get:  host:~ user$ cat example.txt some text host:~ stas$   Is there a way I can make cat behave like this?  I'm using bash on Mac OS X. Unable to Change Apple ID and password I have tried repeatedly to change my Apple ID and password without success. I have followed all of the directions trying both my emails and asked for new passwords----no luck whatsoever. What is Kirito's relationship with the girls of SAO? What differentiates the scientific method from other methods From the controversy in , I would like to know what differentiates "the scientific method" from other method.  I'm asking for differentiation, rather than a definition, because definitions call for an all encompassing answer, which is often hard to find.  If someone asks "what is an alternative to the scientific method," I would like to be able to start by being able to reliably say whether a particular method is the scientific method or not.  Of course, being a linguistic issue, the definitions of words themselves are flexible.  For instance, it is not uncommon to hear one in science talk of "truth" in a sense which is incompatable with the mathematical meaning of "truth."  Accordingly, I would like to stick to the following ideals in wording whenever possible:   Prefer mathematically accurate terminology.  Ideally the word "proof" should be used in the form found in proof theory, and "true" and "false" should be used in the form of Boolean algebra.   No method should claim a hypothesis which predicts the state of the future is "true" unless they claim soothsaying abilities (most scientific theories include a random variable to avoid needing to claim soothsaying abilities... this is legitimate)  Words which imply direction should be clarified carefully.  "Good" and "bad" are notoriously difficult, but phrases such as "advancing society" can also be very tricky as well. The physical world and the world of information theory should be treated differently.  In particular, the concept of "testing a hypothesis" will likely need to be broken down into testing and observing the physical world, taking measurements (which should have errors associated with them), and only then is the hypothesis compared against the results of the measurements. Mathematically extending arguments into infinity (such as claiming some specific cellular level learning process qualifies as the scientific method) are valid, but may rapidly make it harder to distinguish non-scientific methods from scientific ones.  Such claims should be made carefully, but they are not inherently invalid.   If these ideals are too strict, you may bend them, but you should qualify such choices with an explanation of why my recommended wording choices are too strict.  Their goal is to put different methodologies on the levelest playing fields that I can manage in this medium.  Being stack exchange, bonus points for references to existing theories, though you may need to paraphrase their results to fit with the ideals I'm recommending for these answers.  Answers may be tested two ways:   Positive: If we think something is the scientific method, does it meet the requirements set forth? Negative: If we think something is not the scientific method, does the answer explain why it is not? This plant was found growing in Germany. I have never seen it before, reverse Google image search did not help either. Can you help me identify what type of plant this is?      Thanks, looks like it is something like this:    Related question: