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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 $P(0)=P_0$
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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 $n\in\mathbb{Z}.$
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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