0
votes
1answer
26 views

How do I integrate this in terms of error function

How do I evaluate $$\dfrac{1}{\sqrt{4\pi t}}\int_0^{\infty}ye^{-\frac{(\xi-y)^2}{4t}}dy$$ in terms of $\text{erf}(x)$ ? I tried integration by parts but the integral seems to get complicated. I think ...
5
votes
1answer
56 views

How to evaluate $\int_0^ \infty e^{-x\sinh(t)-\frac{1}{2}t}~dt$?

$$ \int_0^ \infty e^{-x\sinh(t)-\frac{1}{2}t}~dt $$ I tried doing it by parts and looking for differentials but I just keep getting back to the original expression. I can't think of a clever ...
0
votes
1answer
62 views

Integral of $\sin|x|$

$$\int\sin|x|~dx$$ We have two cases: x less than zero, or x equals or higher than zero. $$\int_{-\infty}^0\sin(-x)~dx+\int_0^\infty\sin x~dx$$ Left side of this sum is equals to right side, so we ...
1
vote
1answer
27 views

Bessel's integral, how to actually evaluate?

I am just about to study Bessel functions and I have recently seen one of its integral representations given by: $$ J_ \alpha (x) = \frac{1}{\pi} \int_0 ^ \pi \cos(\alpha \tau - x\sin\tau) d\tau - ...
1
vote
1answer
26 views

Provide examples that satisfy the following cases

Provide examples that satisfy the following cases 1) $f_n: [0, ∞)$ → R that converges uniformly to the function $f (x) = 0$ on [0, ∞) but such that $\lim_{n→∞} \int_{0}^{∞} f_n(x) dx \neq ...
1
vote
1answer
107 views

Difficult Improper Integral

Evaluate the improper integral $$\int_0^\infty\frac{-38x}{(2x^2+9)(3x^2+4)} dx $$ I thought about doing this through partial fractions decomposition. However, when I tried, I got some really ...
10
votes
1answer
88 views

Why is an equation necessarily dimensionally correct?

I have just read a fascinating proof of the value of the integral $$ \int_{-\infty}^\infty e^{-ax^2} dx, $$ which proceeds by dimensional analysis, as follows: we know that we can write $$ ...
1
vote
1answer
55 views

Integrating this complex function, using Residue Theorem [duplicate]

I am having a massive amount of trouble integrating this, I really have no clue how to get the answer in the book: $$\int_{-\infty}^{\infty} \frac{x^4}{1+x^8}dx$$ I know I need to find the poles ...
6
votes
4answers
262 views

Does the improper integral exist?

I need to find a continuous and bounded function $\mathrm{f}(x)$ such that the limit $$ \lim_{T\to\infty} \frac{1}{T}\, \int_0^T \mathrm{f}(x)~\mathrm{d}x$$ doesn't exist. I thought about ...
0
votes
1answer
30 views

Show the improper integral $\int^{\infty}_0 \frac 1 y e^{-y} dy$ doesn't converge.

Show the improper integral $\int^{\infty}_0 \frac 1 y e^{-y} dy$ doesn't converge. Using Wolfram Alpha: http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605 the result ...
0
votes
1answer
50 views

Help with finding the definite integral of $e^{\frac{2x-x^2}{2}}$?

I have this integral that I am trying to evaluate by hand, but I am encountering some difficulties. According to Wolfram Alpha, the answer seems to be: However, I do not understand how they got ...
1
vote
1answer
56 views

The Cantor set and integrability of $\frac{1}{x}$

Let $\chi_C$ be the characteristic function of the standard Cantor set fully contained in the interval $[0,1]$. The problem is to resolve if $\lim\limits_{\varepsilon\to 0^{+}} ...
0
votes
1answer
48 views

Calculus 2 Integral Question

I've been trying to resolve a calculus question and seem to be having troubles understanding exactly how to approach it. Some hints are supplied, but they don't exactly seem to help. Thanks to anyone ...
0
votes
2answers
44 views

Evaluating improper integral

Im trying to evaluate the improper integral $$\int_{0}^{\infty}\left( \frac{e^{i \omega t}+e^{-i \omega t}}{2}\right) e^{-st} dt$$, where $\omega$ and $s$ are real positive constants and ...
2
votes
3answers
69 views

Integral $\int_{-\infty}^\infty \frac{dx}{(x^2+a^2)^b}$

For Quantum Mechanics I need to normalize two wavefunctions and I haven't been able to figure out the integrals. The integrals involve a normalization constant, and 2 flexible parameters since these ...
1
vote
1answer
26 views

Show that an integral can be made as small as possible.

Consider a function $\mu(s)$ satisfying the following properties: $\mu(s) \in C^0((0,+\infty))$, $\mu(s) > 0$ and $\mu(s)$ is increasing in $s \in (0,+\infty)$, $\displaystyle \int_0^1 ...
8
votes
1answer
163 views

Closed form for $\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi}\,\sqrt{\sin\phi+\cos\phi}}d\phi$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi}\,\sqrt{\sin\phi+\cos\phi}}d\phi$$ Its approximate numeric value is ...
0
votes
1answer
62 views

$x\rightarrow \int_{0}^{x} \frac{\operatorname{sin}(t)}{t}$ is a bounded function

I've already proved that the improper integral $\int_{0}^{\infty}\frac{\operatorname{sin}(t)}{t}$ is convergent. I don't know its limit though... I'm asked to prove that $\begin{array}{ccccc} ...
1
vote
1answer
58 views

Calculate $\int_0 ^\infty\int_0 ^\infty e^{-x^2 -y^2} dx dy$

I want to calculate the following double integral: $$\int_{0}^{\infty}\int_{0}^{\infty} e^{-x^2 -y^2} \ \mathrm{d}x \ \mathrm{d}y$$ I used the change of variable $x=r\cos\theta$, $y=r\sinθ$, so I ...
1
vote
1answer
38 views

Integral from inverse Fouriertransform of 1/(1+p^2)^2

In a calculation I end up with the following integral $$\int_0 ^\infty \frac{p \sin (pr)}{(1+p^2)^2}dp , $$ could someone give me a hint how to evaluate this one? (This integral comes from the ...
0
votes
0answers
64 views

Different way to see that $\int_{-c}^c \text{d}{x}/x=0$?

The other day, I stumbled across the following integral: \begin{equation} \int_{-c}^c\text{d}{x}\frac{1}{x} \end{equation} with $c$, a positive real number. Now, it seems to me obvious that this ...
4
votes
3answers
95 views

Showing $\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \log(\cos(\phi))\cos(\phi) \ d\phi = \log(4) - 2 $

This is a minor detail of a proof in 'Chaotic Billiards' by Chernov and Markarian which I foolishly decided to verify. It's page 44 of the book, during the proof that lyapunov exponents exist almost ...
1
vote
1answer
24 views

Definite Integral of bessel function of first kind of order one.

How to prove $\int\limits_0^\infty J_1(x)~dx=1$ ? I got $\int\limits_0^\infty J_1(x)dx=-[J_0(x)]_0^\infty$ . Please help.
0
votes
0answers
21 views

Incomplete gamma function and hypergeometric function to Meijer-G

can somebody help me to convert the incomplete gamma function and the hypergeometric function (in the forms shown below and as a function of z) into a form of Meijer-G function?
0
votes
1answer
45 views

Convergence of the infinite series $2^x\ln(1+1/3^x)$

The Q: determine whether the series converges or not $$\sum_{k=1}^\infty 2^k\ln(1+1/3^k) $$ So far I figured out that the function is positive and decreasing on [1,infinity). I decided to try ...
1
vote
3answers
49 views

search for closed form solution of definite integral

Integrate/hint for this definite integral $$\int_0^\infty(\log\theta)^n\frac{1}{\theta^{k+2}}\text{d}\theta,$$ where $n$ and $k$ are positive integers. It is a simplified form of my earlier question ...
8
votes
0answers
80 views

Need help with $\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$

Please help me to evaluate this integral: $$\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$$
1
vote
2answers
72 views

a question about improper integral, I cannot solve it!

If $f(x)$ is continuous in $[0,\infty)$, and $\displaystyle\int_c^{\infty}\frac{f(x)}{x}\, dx$ is convergent for any $c>0$. Please prove $\displaystyle\int_0^{\infty}\frac{f(\alpha x)-f(\beta ...
1
vote
2answers
43 views

Can somebody help me solve the proof about improper integrals?

If $f(x) > 0$ is continuous at $[0, +\infty]$ and $\displaystyle \int_0^{+\infty} \frac{1}{f(x)} dx$ is convergent, please prove $\displaystyle \lim_{\lambda \to \infty} \frac{1}{\lambda} ...
1
vote
1answer
50 views

Length of a Curve and Integration

Two different integration questions are baffling me right now, and I have no idea how to approach them: The first deals with finding the length of the curve $$ y = \int_{-2}^x sqrt(3t^4-1)dt$$ ...
1
vote
0answers
37 views

solve a non-linear integral equation by python

I need to solve an integral equation by python 3.2 in win7. I want to find an initial guess solution first and then use "fsolve()" to solve it in python. This is the code: ...
2
votes
0answers
32 views

Which substitution do I use?

For $x>0$, let $f(x)=\int_0^\infty e^{-t-x^2/t} t^{-1/2}dt$. a) show that $f(x)=x\int_0^\infty e^{-t-x^2/t} t^{-3/2}dt$ via an adequate substitution. b) Calculate $f'(x)$ and show that ...
5
votes
1answer
56 views

Improper Integral $\int_0^2 \frac{1}{\sqrt{x}} \ \text{d}x$

I have to find the value of: $$\int_0^2 \dfrac{1}{\sqrt{x}} \ \text{d}x$$ Here is my work so far: $$\int_0^2 \dfrac{1}{\sqrt{x}} \ \text{d}x$$ $$=\int_0^2 x^{-1/2} \ \text{d}x$$ ...
1
vote
0answers
55 views

How to integrate the following function?

Let's have the integral $$ I(\mathbf r, \omega) = \int \limits_{-\infty}^{\infty} e^{i(\mathbf k \cdot \mathbf r )}\frac{\sin(\omega \sqrt{\kappa^2 + k^2})}{\sqrt{\kappa^{2} + ...
2
votes
1answer
59 views

The antiderivative of $\sin(1/x)$

How to prove that the function $f(x)=\sin\frac{1}{x}$ for $x\neq 0,f(0)=0$ has an antiderivative? This means $F(x)=\int^{x}_{0}\sin(1/t)dt$ has derivative $0$ at $x=0$, but I have no idea how to prove ...
5
votes
6answers
303 views

How to integrate $\displaystyle 1-e^{-1/x^2}$?

How to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: ...
1
vote
2answers
47 views

Can an integral be proved to have a finite value if an upper bound of the integrand has a finite value for improper integrals?

Can we say $ \int_{0}^{\infty} f(x) \text{dx} < \infty$ if $\exists \quad g(x) : \quad g(x)\geq f(x)\; \forall x \in \mathbb{R}$ and $ \int_{0}^{\infty} g(x) \text{dx}$ is finite. If yes, ...
2
votes
1answer
61 views

Magical test for convergence of improper integrals?

I found this article while surfing the web. I hope it's not some kind of joke, because if it is it really fooled me. I'm trying to figure out the proof of theorem 2.3 I don't understand how the ...
1
vote
2answers
90 views

More difficult Integral

This is sort of follow up to my previous question (less difficult integral) here. How do I find $$\large\int_{0}^{\infty}x^ne^{-\left(ax+\frac{b}{x}\right)}dx$$ where $a$ and $b$ are positive reals, ...
0
votes
0answers
34 views

Help with taylor series as part of an integral involving gamma function

I am facing some strange problem regarding the Taylor series for this function: $$\frac{1}{(1+(\eta z)^n)^p} = ...
9
votes
2answers
385 views

How prove this integral $\int_{0}^{\infty}f^{\alpha}(x)dx,\alpha>1$ is convergent

Question: let the function $f(x)\ge 0$,and such $$f'(x)\le\dfrac{1}{2},\forall x\ge 0$$ and this integral $\displaystyle\int_{0}^{\infty}f(x)dx$ is convergent. show that: ...
0
votes
2answers
64 views

Integral test for convergence?

Why does the series' terms have to be non-negative to use the integral test? Consider the series: $$\sum_{n = 1}^{\infty}\frac{n\cos n - \sin n}{n^2}$$ Even though it has negative terms, why can't ...
0
votes
1answer
27 views

Lifetime of exponential variable of a battery

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with parameter $\theta=2$ $($measured in years$)$. Find the probability that a battery of this type ...
1
vote
0answers
30 views

Lifetime of pdf disk

The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows: $f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ...
4
votes
3answers
67 views

What is $\int \sec^3 x \ dx$? [duplicate]

I have to solve for the integral: $$\int \sec^3 x \ dx$$ I used integration by parts, letting: $$u=\sec x$$ $$du = \sec x \tan x \ dx$$ $$dv=\sec^2 x$$ $$v=\tan x$$ Integration by parts formula: ...
25
votes
2answers
463 views

Integral $\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx$

I'm struggling with this integral $$I=\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx.\tag1$$ Mathematica could not evaluate it in a closed form. Its numeric value is ...
2
votes
1answer
43 views

Is this improper integral answer correct?

So I'm working on improper integrals and con/divergence and want some assurance that I've done the following correctly. $\int^∞_{-∞}cos(\pi t)$ As far as I'm aware this is convergent if and only if ...
3
votes
1answer
58 views

Solution of definite integrals involving incomplete Gamma function

The solution of the integral $$\int_0^{\infty}e^{-\beta x}\gamma(\nu,\alpha \sqrt x)dx $$ is given as ...
0
votes
3answers
81 views

Mean value over an infinite interval

I just need a sanity check, been thinking about this all morning. If we use the Mean Value Theorem on a function over the infinite interval (suppose the function's domain is unbounded), i.e. ...
17
votes
4answers
213 views

Integral $\int_0^1\frac{\log(1-x)}{\sqrt{x-x^3}}dx$

I have a trouble with this integral $$I=\int_0^1\frac{\log(1-x)}{\sqrt{x-x^3}}dx.$$ Could you suggest how to evaluate it?