2
votes
2answers
74 views

Integral of ln(x)sech(x)

How can I prove that: $$\int_{0}^{\infty}\ln(x)\,\mbox{sech}(x)\,dx=\int_{0}^{\infty}\frac{2\ln(x)}{e^x+e^{-x}}\,dx\\=\pi\ln2+\frac{3}{2}\pi\ln(\pi)-2\pi\ln\!\Gamma(1/4)\approx-0.5208856126\!\dots$$ I ...
0
votes
0answers
59 views

Mathematica Integrate gives back the integrand

i'm trying to Integrate the following function: (q (1 + q) - E^-q Sinh[q])/(-q + Cosh[q] Sinh[q]) - ( 2 q Tanh[q])/(-q + Cosh[q] Sinh[q]) I already solved ...
5
votes
5answers
185 views

Evaluating the following integral: $ \int \frac{x^2}{\sqrt{x^2 - 1}} \text{ d}x$

For this indefinite integral, I decided to use the substitution $x = \cosh u$ and I've ended up with a $| \sinh u |$ term in the denominator which I'm unsure about dealing with: $$\int ...
3
votes
1answer
70 views

given $\cosh u = x$ find $\sinh u$

I'm asked to show that:$\newcommand{\arcosh}{\operatorname{arcosh}}$ $\int{x \arcosh x}dx = \frac{1}{4}(2x^2 -1)\arcosh x - \frac{1}{4}x\sqrt{x^2 -1} + C$ If I integrate by parts: let $u = \arcosh ...
1
vote
1answer
35 views

Disappearing negative signs when evaluating a sinh^-1 integral

$$\int_{-2}^{6} \frac{1}{\sqrt{1+(-x)^2}} \, dx$$ When performing this integral on paper, I get $$\sinh^{-1}(6) - \sinh^{-1}(-2) $$ But when I type it on wolframalpha, I get the unintuitive answer ...
0
votes
2answers
85 views

I cannot find the following integral in an integral table.

In the appendix A of this paper there is an integral that the author says can be solved using any good integral table. However I cannot seem to find it on any integral table (ex: gradshteyn and ...
1
vote
1answer
89 views

How do I solve this integral with hyperbolic functions?

I was studying mechanics when I f ound a problem that lead to an integral that I can't solve. Basically the problem asked to find the period of oscillation function of the energy $E$ of a particle ...
5
votes
1answer
112 views

Weierstrass $ \tanh \frac{\theta}{2} $ substitution confusion.

I'm already familiar with the trigonometric version of this substitution $ t = \tan \frac{\theta}{2} $ and it's geometrical derivation involving the unit circle found here. However, I'm not sure how ...
0
votes
1answer
47 views

find $\int{artanh(x)} dx$

I've been asked to find $\int{artanh(x)} dx$ The first thing I did was I said: let $y = artanh(x)$ $\therefore \dfrac{dy}{dx} = \dfrac{1}{1-x^2}$ $\therefore dx = (1-x^2)dy$ Also, from $y = ...
0
votes
2answers
41 views

Questions about hyperbolas and integration

I have a couple of questions regarding hyperbolas and their integrals. If it's too much, don't feel like you have to answer all 3 questions. My first question: The integral of a function like 1/x^2 ...
0
votes
1answer
66 views

How to find the angle subtended to the origin by the unit hyperbola through the point (1,0)?

I'm trying to find the angle subtended by the unit hyperbola through the point (1,0). I think that I should be integrating something, but I'm not sure how to set it up. I've been trying to think of ...
3
votes
0answers
79 views

Closed form for $\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$

Find a closed form for $$\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$$ What I have tried Expanding the $\mathrm{arccsch}$ into its logarithmic form, however I ...
3
votes
3answers
163 views

Find the limit of $S_n=\sum_{i=1}^n \big\{ \cosh\big(\!\!\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$, as $n\to\infty$?

$S_n=\sum_{i=1}^n\big\{ \cosh\big(\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$ as $n\to\infty$ I stumbled on this question as an reading about Riemannian sums as in $$ \int_a^b f(x)\,dx =\lim_{x\to ...
0
votes
2answers
79 views

How to integrate this $\frac{1}{(1-e^{2x})^{1/2}}$?

Please how to integrate this $$\frac{1}{(1-e^{2x})^{1/2}}$$ I have tried $u= e^x$ But I think that is wrong So can anyone help me ?
2
votes
3answers
80 views

What is wrong? Symmetric function

I need some advice here. What is $y(\ln(4))$ if the function $y$ satisfies: $$\frac{dy}{dx} = 1-y^2$$ and is symmetric about the point $(\ln(9),0)$. Solving that equation I end up with: ...
47
votes
2answers
1k views

Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx$

Consider the following integral: $$\mathcal{I}=\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx,$$ where ...
6
votes
1answer
184 views

A conjecture $\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx\stackrel?=\frac\pi2\ln2$

I need to find a closed form for this integral: $$\mathcal{I}=\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx.$$ A numerical integration results in an approximation ...
4
votes
1answer
301 views

fourier expansion of $\coth$ and justifying an identity

The problem: Justify the following equalities: $$\cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2}$$ I am trying to figure ...
2
votes
0answers
48 views

Integral with $\cosh$ and $\log$ in the integrand

I am trying to find a good way to simplify (or even solve) the following integral: $$ \int_0^{-\log \varepsilon} \cosh^3r \frac{\partial^2}{\partial r^2} \log (\det E) dr $$ where $r > 0$ is a ...
13
votes
3answers
284 views

Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$

Let $$f(a)=\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx,$$ where $\operatorname{sech}(z)=\frac2{e^z+e^{-z}}$ is the hyperbolic secant. Here are values of $f(a)$ at some ...
1
vote
1answer
119 views

Integral of $\operatorname{cosech}^4 x$

$\newcommand{\cosech}{\operatorname{cosech}}$ I ran into this integral when trying to solve a problem, and I could not get my head around it. $$\int \cosech^4(x)dx$$ I tried to split into two ...
1
vote
0answers
84 views

Solution by of nonlinear equation

$$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \sin u = 0$$ From the sine-Gordon equation we can easily solve, \begin{equation} \phi(x) = \pm 4 \tan^{-1}\left[e^{\frac{x-t ...
3
votes
0answers
235 views

Integrating a fractional power of a rational function

I am currently working on a project where I stumbled upon the integral $$ \int \frac{\sinh \left(\frac{R}{2}\right)}{(\coth R - 6R \coth\left(\frac{R}{2}\right) + 9)^{1/4}} \,dR $$ where $R$ is a ...
1
vote
1answer
54 views

Hyperbolic integration solving

$$ \therefore x-x_0 = \pm \int_{\phi(x_0)}^{\phi(x)} \frac{d \phi}{\sqrt\frac{\lambda}{2}\left( \phi^2-(\frac{m}{\sqrt \lambda})^2\right)} $$ How can we write the above equation to as, $$ \phi(x) = ...
2
votes
1answer
540 views

Integration Using Hyperbolic Substitution

I'm supposed to prove the following by using the hyperbolic sine double angle identity: $\sinh(2x)=2\sinh x\cosh x$ and archsinhx formula: $\ln(x+\sqrt{x^2+1})$ but can't seem to figure out the steps. ...
2
votes
1answer
80 views

definite Integration

A integration is given, $$M = \int_{- \infty}^{\infty} \left[\frac{1}{2} \left(\frac{d\phi}{dx}\right)^2 + \frac{\lambda}{4}(\phi^2-v^2)^2\right] dx,$$ where $$m=v\sqrt\lambda$$ and $$ \phi(x)= ...
2
votes
1answer
145 views

Evaluation of an integral involving hyperbolic sine and exponential

I am wondering if the following integral can be reduced to either a closed form involving elementary functions, or well-known special functions (such as $\operatorname{erf}$, Bessel functions, etc.): ...
0
votes
1answer
196 views

Find area of a curvilinear triangle that includes hyperbolic functions

We were given this question in class and I tried to compute it and it looks to be pretty crazy. Can anyone take a look and let me know if I did it correctly? I would really appreciate it. ...
0
votes
1answer
129 views

Integration by parts

Question 2)b) part (ii) is the section that I'm having trouble with: I don't understand the method used in the solutions; how would you deduce the first line or is that something you should know? ...
3
votes
1answer
164 views

Evaluation of integral involving $ \tanh(ax) $

Is it possible to evaluate in closed form the integral $$ \int_{-\sqrt{x}}^{\sqrt{x}}\frac{r\tanh(ar)}{\sqrt{x-r^{2}}}dr=2\int_{0}^{\sqrt{x}}\frac{r\tanh(ar)}{\sqrt{x-r^{2}}}dr$$ here $a$ is a ...
0
votes
1answer
38 views

For the previous question on hypergeo function

For $\displaystyle\int_0^{\infty}x^n\frac{\sinh ax}{\cosh bx}\text{d}x$, $\left|a\right|<b$ I think we can evaluate $\displaystyle\int_{0}^{\infty }{{{\text{e}}^{cx}}\frac{\sinh ax}{\cosh ...
2
votes
1answer
317 views

A hard integral with hyperbolic function

I was self studying integral. I meet a difficult problem here: $$\int_{0}^{\infty }{{{x}^{n}}\frac{\sinh ax}{\cosh bx}}\text{d}x=\frac{\pi }{2b}\cdot \frac{{{\text{d}}^{n}}}{\text{d}{{a}^{n}}}\tan ...
2
votes
1answer
72 views

Some estimate concerning hyperbolic functions

I want to show that $|\sinh(az)|\leq|\sinh(z)|$ for all $z\in\mathbf{C}$ (or at least for all $z\in\mathbf{H}$, the upper half plane), provided that $0<a<1$ However, I am not even certain ...
5
votes
1answer
149 views

Integral with hyperbolic cosine squared

Does anyone can give me a hint how to integrate the following: $$\int_0^\infty{\frac{x^2 {\rm d}x}{\mathrm{cosh}^2(x)}}.$$ The answer is $\frac{\pi^2}{12}$ (taken from the book). I've started with ...