Michael Hardy
Reputation
144,226
99/100 score
 2d reviewed Edit Evaluate $\int \sin(3x)\cos(4x) \,dx$ 2d revised Evaluate $\int \sin(3x)\cos(4x) \,dx$ MathJax 2d revised Is this well known? added 7 characters in body Apr 28 revised What is the domain of the successor function? edited title Apr 28 comment Finding a Lyapunov function for $u''+u'+\sin u = 0$ Please: Write $\sin u$, not $sin u$. (See my edit to the question for standard MathJax usage.) $\qquad$ Apr 28 revised Finding a Lyapunov function for $u''+u'+\sin u = 0$ added 2 characters in body; edited title Apr 28 comment Indefinite INTEGRAL fraction $\ldots\,{}$Possibly skipping that first step and going straight to the substitution is better. $\qquad$ Apr 28 comment Indefinite INTEGRAL fraction One thought is that if $u=x^2+2x-3$ then $\dfrac{du} 2 = (x+1)\,dx$ and you can write $$\int \frac{x-1}{\cdots\cdots} \, dx = \int \frac{x+1}{\cdots\cdots}\,dx + \int \frac{-2}{\cdots\cdots}\,dx$$and then use the substitution in the first integral. That still leaves the second integral to be done differently. But of course you instantly think of completing the square: $x^2+2x-3 = (x+1)^2 - 4$ and then you can write $x+1 = 2\sec\theta$ and $dx = 2\sec\theta\tan\theta\,d\theta$. I'm not sure where this goes after that; otherwise I'd post an answer. $\qquad$ Apr 28 comment Solve the initial value problem $\frac{dP}{dt}=P(1-\frac{P}{K})$ With $P(0)=P_0$ @user2250537 : I had a minus sign where a plus sign should have been; now I've fixed that. $\qquad$ Apr 28 revised Solve the initial value problem $\frac{dP}{dt}=P(1-\frac{P}{K})$ With $P(0)=P_0$ edited body Apr 28 revised Solve the initial value problem $\frac{dP}{dt}=P(1-\frac{P}{K})$ With $P(0)=P_0$ added 6 characters in body Apr 28 answered Solve the initial value problem $\frac{dP}{dt}=P(1-\frac{P}{K})$ With $P(0)=P_0$ Apr 28 comment Solve the initial value problem $\frac{dP}{dt}=P(1-\frac{P}{K})$ With $P(0)=P_0$ You should probably make your answer posted below into a part of the question above. $\qquad$ Apr 28 revised Solve the initial value problem $\frac{dP}{dt}=P(1-\frac{P}{K})$ With $P(0)=P_0$ added 16 characters in body Apr 28 revised Find minimal possible value of the expression $4\cos^2\frac{n\pi}{9}+\sqrt[3]{7-12\cos^2\frac{n\pi}{9}},$ where $n\in\mathbb{Z}.$ edited body Apr 28 comment Proof that there are infinitely many primes (Euclid) @M47145 : That has been my understanding. See this page: aleph0.clarku.edu/~djoyce/elements/bookVII/bookVII.html $\qquad$ Apr 28 comment Proof that there are infinitely many primes (Euclid) In the context of Euclid, $1$ is not a number, so "No number divides $1$" would be correct. Apr 28 comment Proof that there are infinitely many primes (Euclid) @ASKASK : Finiteness of the arbitrary finite set of primes plays a role. An assumption of finiteness of the set of ALL primes plays no role in the construction, even if it plays a different role of the kind you suggest. Apr 28 comment Proof that there are infinitely many primes (Euclid) @ASKASK : If you want to look at it that way, it is still better not to create an illusion that the assumption of finiteness of the set of ALL primes somehow plays a role in the construction. Apr 28 revised $A,B\in\mathbb Q[x]$ with $A,B$ monic, and $AB\in\mathbb Z[x]$, prove $A,B\in\mathbb Z[x]$ added 2 characters in body; edited title