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 Nov 12 awarded Nice Answer Nov 12 comment What is the integral of $x(1-x)^8$? Why $x=(x-1)+1$? Because I want to treat $x-1$ as sort of a unit, which makes the calculation easier. Nov 12 comment What is the integral of $x(1-x)^8$? $(x-1+1)(x-1)^8=(x-1)\cdot (x-1)^8+1\cdot (x-1)^8$ follows from the distributive law. Nov 12 revised What is the integral of $x(1-x)^8$? added 112 characters in body Nov 12 answered What is the integral of $x(1-x)^8$? Nov 11 comment Finding pairs of integers such that $x^2+3y$ and $y^2+3x$ are both perfect squares Symmetry ($x^2+3y$ and $y^2+3x$) suggests that without loss of generality we can assume $y\ge x$. Also notice that if $x\sim y$, $y^2+3x$ is not much larger than $y^2$, so it may be equal to $(y+1)^2$ or $(y+2)^2$... To discuss this rigorously you need to assume $y\ge x$. Nov 10 comment Show that the value of a definite integral is unity As Dinesh points out, $g(x)=\frac{f(x)}{f(x)+f(6-x)}$ satisfies $g(x)+g(6-x)=1$ (more generally, $g(x)+g(a-x)=b$ where $a$ and $b$ are constants); this is the condition where you can use this integration trick. Nov 10 comment Is $x^2+ax+a$ irreducible over ring $\mathbb{Z}$ of integers? @pedja I think in that case, the same method as posted by Bruno Joyal works (i.e. think in $\mathbb{F}_2[x]$). Nov 10 answered Is $x^2+ax+a$ irreducible over ring $\mathbb{Z}$ of integers? Nov 10 comment Must every event have a probability? The probability that you can find life on Mars (or a Nessie in Loch Ness). I don't think it is "impossible to define any probability", but I think it is mathematically ill-defined (is it?). Nov 10 revised Show that the value of a definite integral is unity edited body Nov 10 revised Show that the value of a definite integral is unity added 337 characters in body Nov 10 comment Show that the value of a definite integral is unity Because $y$ is a dummy variable; please see my edited comment. Nov 10 revised Show that the value of a definite integral is unity added 111 characters in body Nov 10 comment Show that the value of a definite integral is unity I posted an answer below. Nov 10 answered Show that the value of a definite integral is unity Nov 10 comment Show that the value of a definite integral is unity In general: $\int_2^4 \frac{f(x)}{f(x)+f(6-x)}\,dx=1$. Nov 9 awarded Enlightened Nov 9 revised Equivalent conditions to $0\leq x+\frac{1}{2}x(1-x)a\leq 1$? added 12 characters in body Nov 9 comment Equivalent conditions to $0\leq x+\frac{1}{2}x(1-x)a\leq 1$? @JavaMan Not necessarily "opening downwards" -- if $a<0$.