I am trying to find $$\int \frac {\sqrt {x^2 – 4}}{x} dx$$ I make $x = 2 \sec\theta$ $$\int \frac {\sqrt {4(\sec^2 \theta – 1)}}{x} dx$$ $$\int \frac {\sqrt {4\tan^2 \theta}}{x} dx$$ $$\int \frac {2\tan \theta}{x} dx$$ From here I am not too sure what to do but I know I shouldn’t have x. $$\int […]

$y”-2\sin y’+3y=\cos x$ I’m trying to solve it by power series, but I just can’t find the way to get $\sin y’$. Is there any special way to find it?

I’m currently taking a Comp Sci class that is reviewing Calculus 2. I have a question: Show that the summation $\sum_{i=1}^{n}\frac{1}{i^2}$ is bounded above by a constant I realize that this question is already answered here Showing that the sum $\sum_{k=1}^n \frac1{k^2}$ is bounded by a constant Could anyone explain it to me further? I […]

If $\Phi(y)$ is a monotonic decreasing function is true that $$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\Phi'(\Phi^{-1}(y))}$$ If so, how? It works for $y = \Phi(x) = e^{-x}, \quad \Phi^{-1}(y) = -log(y), \quad \frac{d\Phi^{-1}(y)}{dy} = \frac{-1}{y}, \quad $

How do we solve this given $f'(0)=-1$. It does not look separable. I can integrate both sides but end up with a functional equation with is not helpful.

Recall the definitions of the sine and cosine integrals: $$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos t}t dt.$$ Both functions are oscillating, with a countably infinite number of minima and maxima. Note that $$\lim_{x\to\infty}\operatorname{Si}(x)=\frac\pi2,\quad\lim_{x\to\infty}\operatorname{Ci}(x)=0.$$ Consider the following function: $$f(x)=\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}.$$ It appears that the function $f(x)$ and all its derivatives are monotonic for $x>0$. Specifically, the function itself and all […]

I had to integrate $$\int\frac{x^2+1}{(x^2-1)^2} dx$$ Well my first approach was to write$\ (x^2+1)$ as $\ (x^2-1)+2$ so as to obtain fractions $$\frac{1}{(x^2-1)} + \frac{2}{(x^2-1)^2}$$ Now I know how to integrate the first part but how to integrate the second part i.e. a quartic (biquadratic) in the denominator? (I got the answer to the original […]

$$\lim_{x \to \pi/2} \frac{\sqrt[4]{ \sin x} – \sqrt[3]{ \sin x}}{\cos^2x}$$ I have an idea of replacing $\sin x$ to $n$ when $n \to 1$ but wolfram says that answer is $\frac{\pi}{48} $ so my suggestion is it’s had to use trigonometry simplifications which I do not know so well. Assuming that L’Hopital is forbidden but […]

In class I saw a proof that went something along these lines: Define $\|A\| = \sup \dfrac{\|Av\|}{\|v\|}$ for v in V, where the norm used is the standard (Does this even exist?) Euclidean norm in V. $\|Av\|^2 = <Av, Av> = <A^TAv,v>$ where $<,>$ denotes a dot product. Note that $A^TA$ is a non-negative matrix, […]

Does anybody have a proof of the concavity of the $\log{x}$ that does not use calculus?

Intereting Posts

Prove that $\sqrt 2 +\sqrt 3$ is irrational.
The Axiom of Choice and the Cartesian Product.
Why is the following language decidable? $L_{one\ right\ and\ never\ stops}$
The Prime Polynomial : Generating Prime Numbers
Closed form for $\int_1^\infty\frac{\operatorname dx}{\operatorname \Gamma(x)}$
How to find the minimum/maximum distance of a point from elipse
What is a projective ideal?
Why does “the probability of a random natural number being prime” make no sense?
How to represent XOR of two decimal Numbers with Arithmetic Operators
Evaluate the series $\lim\limits_{n \to \infty} \sum\limits_{i=1}^n \frac{n+2}{2(n-1)!}$
Induction for statements with more than one variable.
Multiplicative group $(\mathbb R^*, ×)$ is group but $(\mathbb R, ×)$ is not group, why?
Picard's existence theorem, successive approximations and the global solution
Automorphisms of $k]$ which are also automorphisms of $k$
Traversing the infinite square grid