How to show the following      $$\displaystyle  \int_{0}^{1}\dfrac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}dx  \\ =  \pi\sqrt{\frac{1+\sqrt{13}}{78}} \log \left(\frac{1+\sqrt{13}+\sqrt{2  \sqrt{13}-2}}{4} \right)+\pi\sqrt{\frac{\sqrt{13}-1}{78}}\tan  ^{-1}\left(\frac{\sqrt{5+2 \sqrt{13}}}{3} \right)$$   It seems that it can be transformed into an elliptic integral. Maybe we can define   $$I(a) = \int_{0}^{1}\dfrac{\arcsin{a\sqrt{x}}}{x^4-2x^3+2x^2-x+1}dx $$  Then by differentiation   $$I'(a) = \int_{0}^{1}\dfrac{\sqrt{x}}{(\sqrt{1-a^2x})(x^4-2x^3+2x^2-x+1)}dx $$  Another approach might involve series expansion of $\arcsin(x)$  $$\arcsin(x) = \sum_{n\geq 0}\frac{{2n \choose n} }{4^n(2n+1)}x^{2n+1}$$  To be honest it seems helpless.   Reference
Find this integral $$\operatorname I=\int\limits_{0}^{1}\dfrac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$$  My try: let $$f(x)=x^4-2x^3+2x^2-x+1$$ I found  $$f(1-x)=(1-x)^4-2(1-x)^3+2(1-x)^2-x+1=x^4-2x^3+2x^2-x+1=f(x)$$ so $$I=\int_{0}^{1}\dfrac{\arcsin{\sqrt{(1-x)}}}{x^4-2x^3+2x^2-x+1}dx$$ so $$2I=\int_{0}^{1}\dfrac{\arcsin{\sqrt{x}}+\arcsin{\sqrt{(1-x)}}}{x^4-2x^3+2x^2-x+1}dx$$ then I can't,Thank you very much
Suppose $n$ divides $3^n + 4^n$. Show that $7$ divides $n$. Suppose $n \geq 2$ and $n$ is a divisor of $3^n + 4^n$. Prove that $7$ is a divisor of $n$.  My work so far:  I had a hypothesis that if $n| 3^n + 4^n$, then $n = 7^k$ for some $k\in\mathbb{N}$. But this is not necessarily so. Take $n = 7⋅379$, where $3^7 + 4^7 = 7^2⋅379$. Then, $3^7+4^7$ divides $3^n + 4^n$, and since $n$ divides $3^7+4^7$, we must have $n|3^n+4^n$. Natural Sort in MySQL A doubt from Milnor's "Topology from a Differentiable Viewpoint". This is a doubt from Milnor's "Topology from a Differentiable Viewpoint".     For a smooth $f:M\to N$, with $M$ compact, and a regular value $y\in N$, we define $n(f^{-1}(y))$ to be the number of points in $f^{-1}(y)$. It is easy to see that $n(f^{-1}(y))$ is finite. It can also be seen that $n(f^{-1}(y))$ is locally constant; i.e. there is a neighbourhood of $y$ such that for every point $y'$ in that neighbourhood, $n(f^{-1}(y'))=n(f^{-1}(y))$.   The proof for the statement in bold is the following:     Let $x_1,x_2,\dots,x_n$ be the points of $f^{-1}(y)$, and choose disjoint neighbourhoods $U_1,U_2,\dots,U_n$ of these points, which are mapped diffeomorphically into neighbourhoods $V_1,V_2,\dots,V_n$ of $y$. The required neighbourhood is $(V_1\cap V_2\cap\dots V_n)-f(M-U_1-U_2-\dots U_n)$.   I don't understand how $(V_1\cap V_2\cap\dots V_n)-f(M-U_1-U_2-\dots U_n)$ is open. I know that $(V_1\cap V_2\cap\dots V_n)$ is open. However, how can one prove that $f(M-U_1-U_2-\dots U_n)$ is closed? I can see this working if $f$ is surjective, but not otherwise. drawing at least 90% of colors from urn with large populations How do I change my default account with Google multiple sign-in? After I enable multiple sign-in, and login to multiple accounts, I see that one of them says "(Default)". How do I change which of the accounts is the default one? How should an educational institute woo good faculty? What steps must an educational institution wanting to recruit good faculty take? There is a cycle in the academia:    Good doctorates from top universities invariably join other top universities as faculty members.  Good students aspire to study in top universities under excellent faculty members.   Suppose a new educational institution or department springs up and desires to set up a top-quality group, how does it break into this cycle? I suppose this could take some years, but is money-power the only thing that could be wielded by the institute to achieve its end? How do you keep parents of floated elements from collapsing? I'm going to encrypt the characters Zhu, and decrypt them using RSA. I'm using the public key $\{e, n\}$ and private key $\{d, n\}$. The values of $e$, $d$ and $p$ I get from my textbook: $e = 17$, $d = 53$, $n = 77$.  It works fine when the number is smaller than 76, but in my example below, it doesn't work:  z = 90  h = 104  u = 107  e = 17 d = 53 n = 77  print z,h,u  c_z = z ** e % n c_h = h ** e % n c_u = u ** e % n  print c_z, c_h, c_u  p_z = c_z ** d % n p_h = c_h ** d % n p_u = c_u ** d % n  print p_z, p_h, p_u   The outputs are:  90 104 107 62 69 46 13 27 30     Do you have any idea what's going wrong? Homotopy functions I'm new here. Can anyone give me a hint in this problem? Please.  Let $X$ be a topological space. If $f,g$ are continuous function of $f,g\colon X\to S^n$  which $f(x)\neq -g(x)$ for all $x$ in $X$, show $f$ and $g$ are homotopics. I try to show a homotopy using $$q(x) = \frac{f(x)+g(x)}{\|f(x)+g(x)\|}$$ it's well defined, but i'm stuck. Is there an (easy) way to create or personalize .bst files? what is a thyristor As part of my 2330 lv3 am required to know about thyristors. I just want to confirm my understanding is correct. A thyristor is basically similar to a transistor, only it stays switched on when it's trigged for as long as there isn't a break in the circuit it's switching. Am I correct? Why isn't my website in Google search results? The polymorph spell states, in part:     The target’s game statistics, including mental ability scores, are   replaced by the statistics of the chosen beast. It retains its   alignment and personality.   Does retaining its personality suffice to let it remember who its allies are?  Even if transformed into something stupid like a T-rex?  What does it even mean for a beast with Int 2 to have someone's "personality"? How do I create a 50x3 cell array from a 50x3 matrix. Keep just getting a 1x1 cell array with a 50x3 submatrix? New top bar doesn't show hot threads across Stack Exchange sites Why is there such a big difference between "Size" and "Size on disk"? Free UML drawing library for .NET I have been trying around different ranges of summations:  $$ \sum_{(m,n) \in \mathbb{Z}^2} \frac{m^2 - n^2}{(m^2 + n^2)^2} = 0$$  That's not any good.  What about if we restrict to $m, n \in \mathbb{Z}$ as positive integers.    $$ \sum_{m &gt; 0, n &gt; 0} \frac{m^2 - n^2}{(m^2 + n^2)^2} = 0$$  Now here's an anti-symmetry.  I do not like taking out the symmetry, but perhaps I can ask about:   $$ \sum_{m &gt;  n &gt; 0} \frac{m^2 - n^2}{(m^2 + n^2)^2}  = \; ? \tag{$*$} $$  There doesn't seem to be a change of variables that can work.  And we've used symmetry about as much as we can.  This looks related to:  $$  \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$  Perhaps this other series ($*$) also has a special value. Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$ Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$  I have expanded it this far:  $$\frac{k \cdot n!}{k!(n-k)!} = \frac{n \cdot (n-1)!}{(k-1)!(n-k)!} $$  but then I am stuck from where to go from there. Include Watir test cases in a Continuous Integration system