Do you think we can express the closed form of the integral below in a very nice and short way? As you already know, your opinions weighs much to me, so I need them! Calculate in closed-form $$\int_{1/2}^1 \frac{\arctan\left(\frac{1-x^2}{7 x^2+10x+7}\right)}{1-x^2} \, dx.$$ I'm looking forwar...

In some notes that I am reading there is the following: $$(\delta s)^2=(\delta x)^2+(\delta y)^2 \Rightarrow \left (\frac{\delta s}{\delta x}\right )^2=1+\left (\frac{\delta y}{\delta x}\right )^2$$ When $\delta x \rightarrow 0 $ we get $$(s'(x))^2=1+(y'(x))^2 \Rightarrow s'(x)=\sqrt{1+(f'(x...

Let $E$ be a closed set of real numbers, and $f: E \to \mathbb{R}$ be continuous. I need to show that there exists a continuous function $g: \mathbb{R} \to \mathbb{R}$ such that $g|_{E} = f$. I was given the hint to take $g$ to be linear on each of the intervals of which $\mathbb{R}\backslash E$...

Let $f(x)$ be a real-entire function such that for all $x>0$ we have $f(x) > 0$, $f'(x) > 0$ , $f '' (x) > 0$. And also $0 < D^M f(0) < D^{M-1} f(0)$. Let $0<T<1$ and $n$ a positive integer. Let $g(x,n) = \frac{f(x)}{x^n}$. Let real $x_0(n)$ satisfy $x_0 (n)> 0$ $g ' (x_0(n),n) = 0$ Let $r...