The Intergral of $\sin^4x$ without using reduction formula
Evaluating $\int P(\sin x, \cos x) \text{d}x$
Rendered image looks pixelated When I hit F12 to render an image and zoom in it is quickly pixelated. How can I render a higher res display? I have looked through the settings and couldn't find a solution. How do I prepare best for my first wedding photography event? I am going to shoot a wedding this weekend, and I want to do my best for my friend's big day.    What can I do to prepare for the day in order to minimize failures and mistakes? How do I avoid missing important shots? What backup plan should I have?   I have been shooting for 6 years or so, and I know wedding is one of the hardest thing to do. I have told my friend what I end up producing will be of no match with the work of dedicated wedding photographers, not even the ones that are about to fall out of business.  They are on a tight budget, they said some photos are still better than none, they really must minimize the cost as much as they can. I exaggerated again that my photos will be worse off than the cheapest wedding photographer you can find.  She has been a close friends for years, I want to do everything I can to help her.  Their wedding is at 12:30pm coming Sunday, I have visited the venue last Sunday at 12:30pm to check traffic condition and also the lighting.  The setup is mainly outdoor (no church), the indoor area will not have a good ceiling for bounce. Being at the middle of the day, this can mean some tricky fill flash. I have jot down a list of must-have shots. I am also asking for the seating plan and the event run-down.  Gear:   Canon EOS 60D , EF-S 15-85mm f/3.5-5.6 IS USM Canon EOS 450D (or I can borrow a 550D), EF 50mm f/1.4 Speedlite 580 EXII Speedlite 270 EXII Yongnuo 460 (a cheap but powerful manual flash as backup) Samsung EX-1 f/1.8 point and shoot (backup of a backup...?)   Other lenses that I have but I don't think will be useful:   50mm f/1.8 EF-S 50-250mm f/4.0-5.6 IS II EF-S 18-135mm f/3.5-5.6 IS (I have the better 15-85 on 60D, should I put this on the 450D as a backup allrounder?) EF-S 60mm f/2.8 Macro (should I replace 50mm f/1.4 with this for added macro? You can hardly use f/1.4 under direct sunlight if you want to have fill-flash)   I have two batteries for all cameras (60D 450D and EX1). I have like in total 64GB SD card (I am really inclined to get more, or bring a laptop for on-the-fly backup, or do both). I recently repaired my 580EXII, it has a brand new flash head so the recycle time is as good as it can be, I just need a tonne of batteries. No I don't have external power pack or battery grip for my cameras.  I have a great monopod, but a cheap tripod.  I am happy to bring everything, just putting them in the boot of my car. The carpark is on-site, walking from venue to my car is under 1 minute.  I will have my girlfriend carry the 450D and shoot details / non-portraits.  One concern that I have is fill flash, I really do not have a large surface to bounce off  from. I have shot events before but they are not as dynamic/critical as a wedding.  How do I prepare for differing weather conditions?  If its sunny, fill flash is probably going to be in every single shot. If its cloudy, I probably will turn off the flash. If it rains, my 60D should be fine I have used it in slight rain before, but I really don't know their plan and if they have extendable shelters etc.  What else should I prepare for? Having no experience I want to be as prepared as I can be. Maybe I have overlooked something, maybe there is something else I can do to ensure the shooting will go smoothly. Your help is greatly appreciated! Often more than one person contributes to the community coming up with the final accepted answer to a question. However, it is only possible to accept a single answer. Restricting to one final answer does make sense for the sake of referencing that answer later, but in many cases that answer is the product of the input from several individuals. I would like to be able to decide how much of a bounty goes to different users, or some other mechanism to recognize the contribution of multiple users. Of course I could always upvote their previous answers, but this is really incorrect if their previous answers aren't the complete, final answer I've accepted.  What think you metaSOrs? I have been researching the meaning of positive semi-definite property of correlation or covariance matrices.   I am looking for any information on   Definition of positive semi-definiteness; Its important properties, practical implications; The consequence of having negative determinant, impact on multivariate analysis or simulation results etc. What is the difference between "INNER JOIN" and "OUTER JOIN"? Is it OK to use a forward slash '/' between words other than the obvious 'and/or'? I don't want to get into the proper use of the phrase 'and/or', but rather to investigate the use of the forward slash between other words. Examples:   I have an interview with him/her this afternoon. He has a large binder/notebook.   And even separating just letters:   I connected the i/o ports. (input/ouput) Mark the field as n/a. (not applicable) I did it w/o her help. (without)   It seems as thought the forward slash can be used to indicate an option, display ambiguity, shorten 2 words and shorten 1 word (and others).  Is it OK to use a slash to separate these words/letters (yes, that was intended), or is it better to explicitly say 'his or her'? One more thing, is it correct to 'chain' these phrases together? I'm horrible with examples but here's one:   I don't open letters/mail that aren't/isn't addressed to me. How do you connect to multiple MySQL databases on a single webpage? I have information spread out across a few databases and want to put all the information onto one webpage using PHP.  I was wondering how I can connect to multiple databases on a single PHP webpage.  I know how to connect to a single database using:  $dbh = mysql_connect($hostname, $username, $password)          or die("Unable to connect to MySQL");   However, can I just use multiple "mysql_connect" commands to open the other databases, and how would PHP know what database I want the information pulled from if I do have multiple databases connected. What does "T" mean in C#? I have a VB background and I'm converting to C# for my new job.  I'm also trying to get better at .NET in general.  I've seen the keyword "T" used a lot in samples people post.  What does the "T" mean in C#?  For example:  public class SomeBase&lt;T&gt; where T : SomeBase&lt;T&gt;, new()   What does T do? Why would I want to use it? There are many (duplicate) questions about the acceptance, popularity and history of singular they (and their, them and themself) around here, it even got a tag of its own. If I didn’t miss anything, however, the proper verb form hasn’t been questioned yet.  As we all know, English third person singular pronouns (it, she, he and one, +body), names (Alice, Bob, …) and nouns (student, teacher, …) demand the +(e)s suffix be added to the finite verb form in simple present, where some auxiliary verbs have a “special” form (is &lt; *bes / *ares and has &lt; *haves). All other subjects don’t, including plural third person pronoun they.  When the plural you replaced thou (with thee, thy / thine), the other marked verb form that had remained in English – i.e. suffix +(e)st or +t – vanished, too. The first and second persons only ever appear as pronouns (I, we; you), not names or nouns, so there was no strong inclination to keep the verbal inflection.  The second person precedent would suggest that singular they be used with uninflected verb forms which is how it’s usually encountered in the wild. Assuming that they someday replaces he and she (and maybe it) it would lead to disagreement with the words the pronoun stands in for:        Alice         goes to her   place  by herself.   Bob           goes to his   place  by himself.   Alice and Bob go   to their place  by themselves. – (common)   Alice and Bob go   to their places by themselves. – (separate)                  She  goes to her   place  by herself.   He   goes to his   place  by himself.   They go   to their place  by themselves.   They go   to their places by themselves.                  They ?goes to their place  by themself.   ditto   They  go   to their place  by themselves.   They  go   to their places by themselves.                  They ?go to their place  by ?themselves.   ditto   They  go to their place  by  themselves.   They  go to their places by  themselves.      So why doesn’t singular they afford s on present-tense verbs like all other third person singular subjects do?   I learned about singular they only long after I had been taught English as a second language in school. That’s why it’s still a conscious decision to use it and hence I could easily adapt to use s forms with it, but would that sound and look funny / strange / wrong to native speakers? Among any three consecutive positive integers one is a multiple of 3 Looking for a trilogy published around the early 80s.  Insect-like aliens are invading earth.  Humans' brains are placed in robotic bodies.  The protagonist is a child in one of these bodies.  He has to learn how to control the robot &amp; conserve his battery power.  Once train, he can fight back against the aliens using heat-guns + physical attacks.    The aliens are landing/spawning on sites all over the globe.  Part of the plot is discovering a method to disrupt their staging points.  It was probably young-adult oriented.  If anyone remembers this, thank you. Can I successfully bake previously-frozen bread dough? Logical explanation why exponentiation operation is not commutative and associative Considering Peano axioms we'll define addition, multiplication and exponentiation operations.  We can then prove that addition and multiplication operations are commutative and associative. The proof for addition operation is pretty straightforward but for for multiplication it a bit more difficult. For example, to prove associativity, we have to prove distributivity first.  Moreover, for the definition itself it is not obvious that commutativity should take place, because the definition is not symmetric.  $a \cdot 0 := 0$  $a \cdot S(b) := a \cdot b + a$  Here $S(b)$ is the "next" function from Peano axioms  This definition is not symmetric with respect to $a$ and $b$, so commutivity is not obvious.  For exponentiation operation neither commutivity nor associativity take place. And I am very curious about the logical explanation why that happens  NOTE: The explanation I am looking for is somehow similar to a Abel–Ruffini theorem which explains WHY there is no an algebraic solution for a general polynomial equation Why is the (non-relativistic) stress tensor linear and symmetric? From :     "[...] the stress vector $T$ across a surface will always be a linear function of the surface's normal vector $n$, the unit-length vector that is perpendicular to it. [...] The linear relation between $T$ and $n$ follows from the fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchy’s equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that the stress tensor is symmetric."   What is the more detailed explanation for this, and/or where can I find it? A Hilbert space is, by definition, a complete inner product space. If $(V,|.|)$ is finite dimensional inner product space of dimension $n$ then it is (topologically) isomorphic to $\mathbb{R}^n$ which is of course complete. My instinct here is to say "and therefore, $V$ is also a Hilbert space". I'm not sure about this step however since completeness depends on the norm and the norm depends on the selected inner product (assuming the induced norm $||x|| = (x | x)^{1/2}$ is used). So, must we place some condition on the inner product or is the topological isomorphism between $\mathbb{R}^n$ enough to guarantee that V is Hilbert? Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$ Two independent Poisson process question How to control the mouse in Mac using Python? What would be the easiest way to move the mouse around (and possibly click) using Python on OS X?    This is just for rapid prototyping, it doesn't have to be elegant. Evaluate the integral $\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$ This integral looks a lot like an elliptic integral, but with cubes instead of squares: $$I(a,b)=\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$$ Let's consider $a,b&gt;0$ for now. $$I(a,a)=\int_0^\infty \frac{dx}{x^3+a^3}=\frac{2 \pi}{3 \sqrt{3} a^2}$$ I obtained the general series solution the following way. Choose $a,b$ such that $a \geq b$, then: $$I(a,b)=\frac{1}{a^2} \int_0^\infty \frac{dt}{\sqrt{(t^3+1)(t^3+b^3/a^3)}}=\frac{1}{a^2} I \left(1, \frac{b}{a} \right)$$ $$\frac{b^3}{a^3}=p, \qquad I \left(1, \frac{b}{a} \right)=I_1(p)$$ $$I_1(p)=\int_0^\infty\frac{dt}{\sqrt{(t^3+1)(t^3+p)}}=2 \frac{d}{dp} J(p)$$ $$J(p)=\int_0^\infty\sqrt{\frac{t^3+p}{t^3+1}}dt=\int_0^\infty\sqrt{1+\frac{p-1}{t^3+1}}dt=$$ $$|p-1| \leq 1$$ $$=\sum_{k=0}^\infty \left( \begin{array}( 1/2 \\ ~k \end{array} \right) (p-1)^k \int_0^\infty \frac{dt}{(t^3+1)^k}$$ Now this is the most problematic part. The first integral of this series diverges. However, it's a constant in $p$, so if we differentiate, it formally disappears: $$I_1(p)=2 \sum_{k=1}^\infty \left( \begin{array}( 1/2 \\ ~k \end{array} \right) k (p-1)^{k-1} \int_0^\infty \frac{dt}{(t^3+1)^k}$$ Now, every integral in this series converges. The integtals can be computed using the Beta function, if we substitute: $$t^3=\frac{1}{u}-1$$ Finally, we rewrite: $$I_1(p)=\frac{\Gamma (1/3)}{3 \sqrt{\pi}} \sum_{k=1}^\infty \frac{k^2}{k!^2} \Gamma \left(k- \frac{1}{2}\right) \Gamma \left(k- \frac{1}{3}\right) (1-p)^{k-1}$$ Or, using the Pochhammer symbol:  $$I_1(p)=\frac{2 \pi}{3 \sqrt{3}} \sum_{k=0}^\infty \frac{(k+1)^2}{(k+1)!^2} \left(\frac{1}{2}\right)_k \left(\frac{2}{3}\right)_k (1-p)^k$$  My questions are:  Is the method I used valid (see the 'problematic part')? How to get this series into a Hypergeometric function form? Is there any 'arithmetic-geometric mean'-like transformation (Landen's transformation) for this integral? How to go about finding it?  If the method I used is correct, it can be used for any integral of the form ($m \geq 2$): $$I_m(a,b)=\int_0^\infty \frac{dx}{\sqrt{(x^m+a^m)(x^m+b^m)}}$$ Content type permission to specific users How do I set content permission so that specific content types become visible/usable to the admin user only, while other content types may be visible by other user roles?  After I added new content types, I checked the rebuild permissions.   Content access is working fine but the problem seems to be the fact that the enw content types aren't made visible / accessible to the other roles...