Questions about the evaluation of specific definite integrals.
13
votes
0answers
208 views
$\int_0^1\mathrm{d} u_1 \cdots \int_0^1\mathrm{d} u_n \frac{\delta(1-u_1-\cdots-u_n)}{(u_1+u_2)(u_2+u_3)\cdots(u_{n-1}+u_n)(u_n+u_1)}$
This question grew out of this one: Given an even integer $n\in 2\mathbb{N}$, compute the integral
$$\int_0^1\mathrm{d} u_1 \cdots \int_0^1 \mathrm{d} u_n ...
12
votes
0answers
654 views
Computing ${\int\limits_{0}^{\chi}\int\limits_{0}^{\chi}\int\limits_{0}^{2\pi}\sqrt{1- \cdots} \, d\phi \, d\theta_1 \, d\theta_2}$?
This question led me to this integral I can not solve:
$$
...
8
votes
0answers
292 views
a way of evaluating integrals without doing anything?
The user known as sos440 posted this:
$$\begin{align*} \sum_{n=0}^\infty \frac{r^n}{n!} \int_0^\infty x^n e^{-x} \; dx & = \int_{0}^\infty \sum_{n=0}^\infty \frac{(rx)^n}{n!} e^{-x} \; dx = ...
5
votes
0answers
93 views
Can someone explain this integration trick for log-sine integrals?
I was working on this rather challenging log-sin integral.
$\displaystyle \int_{0}^{2\pi}x^{2}\ln^{2}(2\sin(x/2))dx=\frac{13{\pi}^{5}}{45}$
The upper limit is a waiver from the norm of ...
5
votes
0answers
185 views
Mixed Bessel Function integral $\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\mathrm{d}z$
A tricky integral I have been working on, and probably doesn't have a solution in terms of known functions, is:
$$\int_{0}^{\infty} e^{- \lambda ...
4
votes
0answers
49 views
Good upper bound for $\int_0^1 (1 + 2x)/\sqrt{x + x^3}$
I am trying to obtain an upper and lower estimate for the integral
$$I = \int_0^1 \overbrace{\frac{1}{\sqrt{x^2+1}} \left( \frac{1}{\sqrt{x}} + 2\sqrt{x}\right)}^{\Large f(x)}\,\mathrm{d}x,$$
and an ...
4
votes
0answers
86 views
Mistake in Bartle's proof of Hake's Theorem?
Here is Bartle's proof of Hake's Theorem found in "A Modern Theory of Integration". I think there is a mistake in the highlighted line:
The Theorem: $f:[a,b]\to \mathbb{R}$ is gauge integrable if and ...
4
votes
0answers
109 views
Is there a name or definition for this popular notation?
I'm sorry if this is a silly question. I've done quite a bit of searching and have not found any definition or name for this symbol/usage, despite immense popularity and convenience. The sources I've ...
3
votes
0answers
20 views
fancy about some properties of kernel functions at infinity
Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not:
$1.$ If $K(x,t)$ is bounded ...
3
votes
0answers
73 views
Evaluating the integral $\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\ln|z-x|dx$
I don't know how to deal with this integral:
$$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\ln|z-x|dx,$$
where z is a complex number.
3
votes
0answers
58 views
Simplify the integral with error function
$\newcommand{\erf}{\operatorname{erf}}$
I have the following integral and I need to simplify the solution. I have written first two steps. I don't know what is the value of
$$
\erf(\infty)
$$
I ...
3
votes
0answers
80 views
What is $\int_0^{\infty}\!e^{-x^2}e^{-ae^{bx^2}}\,dx$?
I've been trying without success to evaluate
$$
\int_0^{\infty}\!e^{-x^2}e^{-ae^{bx^2}}\,dx.
$$
It's not in my integral tables. Wolfram online integrator won't do it. It doesn't seem to be amenable to ...
3
votes
0answers
183 views
Finding the volume of a tetrahedron by given vertices.
Please help me with the problem below.
Find the volume of a tetrahedron with vertices:
$O(0,0,0)$, $A(1,2,3)$, $B(-2,1,5)$, $C(3,7,1)$ by using triple integral.
Hint: First find the the equations of ...
3
votes
0answers
115 views
Is there a closed form expression for this integral?
I've been trying to find a closed form expression/series expansion for the following integral without success:
$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
3
votes
0answers
51 views
Hypervolume under the square of an n-simplex
I posted this question a while ago, but since I didn't have much luck I though I'd reformulate it and try again.
Question: What is the general form of the equation that gives the hypervolume under ...
3
votes
0answers
72 views
Is there a closed-form to this integral?
I am trying to evaluate the following integral:
$$\int_a^bx^\alpha \operatorname{Exp}\left[-\left(\frac{x-\delta}{\sigma}\right)^\alpha\right]dx$$
where $0<a<b$, $\delta\leq a$ and ...
3
votes
0answers
262 views
Integration of nontrivial trigonometric functions
First an example which I know how to solve. If we have the following integral
$$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)}dt$$
there is a very practical way to evaluate it by interpreting it as some ...
3
votes
0answers
157 views
Integrating a product of exponential and complementary error function with square-root of variable in the denominator
I need to evaluate
\begin{equation}
\int_a^\infty \mathrm{erfc}\left( \frac{b}{\sqrt{c\cdot h}} \right) e^{-d\cdot h} dh
\end{equation}
where $\mathrm{erfc}(s) = \frac{2}{\sqrt{\pi}} ...
2
votes
0answers
23 views
Setting up proper integral
I can't get a problem to work out the way it is supposed to according to my book.
First, a relevant theorem for the problem:
THEOREM
Suppose $\{V_j; j \in Z\}$ is a multiresolution analysis with ...
2
votes
0answers
66 views
quick integration notation question
What exactly does the base of this integral sign refer to?
$$\int\limits_{[0\to 1]}$$
Is that an arbitrary contour from the point $0$ to the point $1$? (this is complex analysis)
ps. Here's the ...
2
votes
0answers
17 views
Area of $ M=\{[x,y] \in R^2; x (x^2+y^2) < x^2-y^2; x>0 \} $
I started out with expressing $y$ in terms of $x$:
$$
\begin{equation}
\sqrt{\frac{x^2-x^3}{x+1}} <y
\end{equation}
$$
Now I integrate over $x \in (0,1)$ since I've graphed the above expression.
...
2
votes
0answers
26 views
Is this Riemann sum equivalent to this definite integral?
For the following scenario,
The density of people in a 200-ft long stadium during a concert is
given by c(x), where x is the distance, in feet, from the stage.
Find the number of people at the ...
2
votes
0answers
57 views
Existence of closed form solutions on a definite integral.
I need to find the value of the given expression in a closed form
$$\int_0^{\frac{\pi}{2}} \dfrac{\sin x}{ x} dx$$
My effort:
I know that its a standard definite integral called $Si(x)$
$\sin x= ...
2
votes
0answers
44 views
Evaluate $\int_0^\tau \frac{t\sin(t z)}{z\cos(t z)-\sin(tz)}\text{d}t$
I'm trying to evaluate the following definite integral. Mathematica gives me a complicated expression which I think I can simplify, but I was wondering if there was a "nice" way to evaluate it.
...
2
votes
0answers
57 views
Calculating the integral $\int_{0}^{\pi /6}\sqrt{1-\left(\frac{R_s\sin \theta }{C_L}\right)^2} d\theta$
I want to integrate $I=\int\limits_{0}^{\pi /6}{\sqrt{1-{{\left( \frac{{{R}_{s}}\sin \theta }{{{C}_{L}}} \right)}^{2}}}d \theta}$.
I get incomplete elliptic integral $E(z\mid m)$ in the calculation by ...
2
votes
0answers
46 views
Can any one help me normalize this equation? (Modified 3D Gaussian)
$$\exp\left(
- e^{d-sz}
- 2 \left( \frac{z^2}{r^2f^2}+\frac{x^2+y^2}{r^2} \right)
\right)$$
Note if this equation can't be normalized another equation with similar proprieties would also be ...
2
votes
0answers
92 views
Integral representation of Euler's constant
Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$
where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).
This integral was mentioned in Wikipedia as in ...
2
votes
0answers
79 views
Proof that if $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$
I am asking for a self contained proof of this assertion:
If $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$.
The integral in question is the ...
2
votes
0answers
55 views
definit integral of Airy function
How can I evaluate this definite integral
$$ \int_0^\infty \frac{Ai^2(z+a_n)}{z^2}dz $$
where $a_n$ are the zeroes of the Airy function. Need Help please
2
votes
0answers
78 views
Evaluation of these integrals
I need a hint to evaluate the integrals:
$$ \int _{0}^{p}\tanh(\pi k)k\,dk\quad \text{and}\quad \int _{0}^{s-1/2}\tan(\pi k)k\,dk $$
Here $p$ and $s$ are real numbers.
I know I can evaluate them ...
2
votes
0answers
149 views
Surface integral over a rectangle in Cartesian coordinates with singularity inside integration domain
I would like to integrate:
$f(\rho)=\int\limits_S{\frac{1}{R}dS}$ where $R=||\rho-\rho'||_2$ and $\rho'$ represents the distance from the bottom-left corner of the rectangle. i.e. if the integral is ...
2
votes
0answers
128 views
how to calculate the integral
How to calculate the following integral: for positive constants $a_1, \cdots, a_{n+1}, $ and $i>0$
$$
\int_{S^n\bigcap\{u_m\geq 0,\ m=1,\cdots, n+1\}}\left(\sum_{m=1}^{n+1} ...
2
votes
0answers
165 views
Very tricky Fourier transform
I'm trying to evaluate the following integral using complex function theory:
\begin{equation}
...
2
votes
0answers
73 views
analytic evaluation of $\int_{0}^{\infty} \frac{dx}{\mathrm{Bi}(x)}$
So, just out of random curiosity, I'm trying to find an analytic expression for the following definite integral:
$$\int_{0}^{\infty} \frac{dx}{\mathrm{Bi}(x)}$$.
Where $\mathrm{Bi}(x)$ is the Airy ...
2
votes
0answers
504 views
Two Disk/Washer Method Problems (given a diagram)
Given a diagram from Calculus of a Single Variable by Larson and Edward (9th edition):
I am interested in finding the volume of various regions when rotated about various lines. Specifically, I am ...
1
vote
0answers
34 views
What are some definite integration techniques?
Are there any definite integration techniques which I could learn (calc AB student)? I mean techniques which don't require you to find the anti derivative. Thanks!
1
vote
0answers
30 views
Integral of Bessel function of the first kind and exponential function
I would need to know if there's a closed form for the following integral:
$$\int_{0}^{\infty} x^{-1}J_{\frac{1}{2}}(\pi x)J_{\frac{1}{2}}(\pi x)\exp(-b(x-x_0)^2)$$
with $b>0$ and $x_0\in ...
1
vote
0answers
34 views
Integral of gaussian and sine/cosine
I really need the solution of two integrals involving exponentials and sine/cosine.
For $n\in \mathbf{N}$ even :
$$\int_{-\infty}^{+\infty}\left(\frac{2\,ax\sin(\pi ...
1
vote
0answers
19 views
Closed form for $k$-th moment
I would like to calculate this $k$-th moment:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(i^n\frac{\sin(\pi a x+\frac{n\pi}{2})}{\pi ax+\frac{n\pi}{2}}+(-i)^n\frac{\sin(\pi a ...
1
vote
0answers
30 views
Please help finishing the calculation to prove that ” Pareto distribution & Power distribution has inverse relationship”.
Let X follows Pareto distribution with parameters α, a, h.
that is X~Pa(α,a,h)
Where, α>0 is the shape parameter, -∞< a <∞ is the location parameter, h>0 is the scale parameter.
...
1
vote
0answers
40 views
Definite integration of a high order exponential function mixed with rational function
I would like to solve the integral
$$\int_{x>0} x e^{ax^m+bx^n}dx,\qquad m>n>0$$
1
vote
0answers
17 views
Darboux integrable periodic function is integrable on any closed and bounded interval
How do I show that a Darboux integrable periodic function, with period $T$, is integrable on any closed and bounded interval?
Intuitively it is so clear to me, but how one can provide a proof?
...
1
vote
0answers
44 views
Series sum and integral inaccuracy problem
Well, it happened that math is not my field. Having a vague concept in general I have been struggling for a few hours with the exercise. In general, I want to calculate how much interest a man have ...
1
vote
0answers
57 views
Mellin tranform of $\cos x$ using Ramanujan's master theorem
I've been messing around with Ramanujan's master theorem.
$\displaystyle \int_{0}^{\infty} x^{s-1} \cos x \ dx = \int_{0}^{\infty} x^{s-1} {}_0F_1 \left(-;\frac{1}{2};\frac{-x^{2}}{4} \right) \ dx $
...
1
vote
0answers
58 views
Double integral
I want to evaluate
$$ \int_{-\infty}^a \int_{-\infty}^b z_1^n z_2^k \phi(z_1,z_2) dz_1 dz_2$$
where $ \phi(z_1,z_2)$ is a pdf of bivariate normal distribution and $n,k$ are natural numbers. Are there ...
1
vote
0answers
93 views
Evaluating the Gaussian integral $ \int_{-\infty}^{1}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx $ with the substitution $x=\frac{1}{t}$
I got $dx=-t^{-2}dt$, giving me
$$
\int_{0}^{1}-\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2t^{2}}}t^{-2}dt
$$
I can already tell that the above equation can't be right since it is negative everywhere. It ...
1
vote
0answers
59 views
Proof of $f=0$ a.e. in $[a,b]$ then $f$ is gauge integrable and $\int_a^bf=0$
Let $f:[a,b]\to \mathbb{R}$ so that $f(x)=0$ almost everywhere in $[a,b]$. Prove that $f$ is gauge integrable and $\int_a^bf=0$.
How can this be proven using the following definition of measure: ...
1
vote
0answers
42 views
Alternative explanation for $\iint_D \left|\log \left( \frac{e}{1-z} \right) \right|^2 \ dA = \frac{\pi^3}{6}$?
I thought up a curious definite integral. Let $D = \{ z \in \mathbb{C} : |z|<1\}$. Let $A$ denote area measure on $D$, normalized so that $A(D) = \pi$. I claim that $$\iint_D \left|\log \left( ...
1
vote
0answers
53 views
Proof that if $f$ is integrable then so is $\left|f\right|$ with the classical definition of the Riemann Integral
Let $f:[a,b]\to \mathbb{R}$ be integrable. Prove $\left|f\right|$ is integrable with the definition of the integral using the Riemann sums (not the Darboux one).
I think the way to go is to use the ...
1
vote
0answers
85 views
Integrating derivative ($f$ of bounded variation)
Let $f$ be of bounded variation on $[a,b]$, and define $ν(x)=TV(f_{[a,x]}) ∀x∈[a,b]$,
I want to show: $\int_a^b|f'|= TV(f)$ iff $f$ is absolutely continuous on $[a,b]$.
My attempts:
I've shown ...
