Questions on the evaluation of definite and indefinite integrals
2
votes
1answer
48 views
Integral of product of Bessel functions of the first kind
I would like to solve the integral:
$$\int_0^{+\infty}\quad rJ_n(ar)J_n(br)\quad dr$$
Is there any closed form for it?
Thanks!
2
votes
3answers
141 views
How to solve this integral for a hyperbolic bowl?
$$\iint_{s} z dS $$ where S is the surface given by $$z^2=1+x^2+y^2$$ and $1 \leq(z)\leq\sqrt5$ (hyperbolic bowl)
2
votes
3answers
70 views
The indefinite integral $\int \frac{1+\cos(x)}{\sin^2(x)}\,\mathrm dx$
I`m trying to solve this integral and I did the following steps to solve it but don't know how to continue.
$$\int \frac{1+\cos(x)}{\sin^2(x)}\,\mathrm dx$$
$$\begin{align}\int \frac{\mathrm ...
2
votes
2answers
32 views
Integral of $\int \frac{\sin(x)dx}{3-\cos(x)}$
I am trying to solve this integral and I need your suggestions.
I don't know if its OK to set $3-\cos(x)$ as $t$ $\rightarrow dt = \sin(x)dx$ or just take $\cos(x)$ and set it as $t$
$$\int ...
2
votes
2answers
57 views
Complex-valued Fourier integral: $ \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $
I'm working on the Fourier transform, but I don't know how to evaluate the integral:
$$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$
0
votes
2answers
48 views
What's a better way to integrate this?
$$ \int \frac{1}{x^2 + z^2} dx $$
I tried substitution and also by parts. By parts is getting messy and I am not sure if I am getting the right answer. I am trying to figure out an easier way or the ...
-1
votes
0answers
24 views
Flux integrals, parameterization
let S be the cylinder x^2 + z^2 = 9 where -2 /ge y /le 2
parameterization: thi(u,v)= <3cosv, u, 3sinv> where -2 /ge y /le 2 and 0 /ge v /le 2pi
(thi is the symbol of I with the circle in the ...
0
votes
0answers
47 views
Evaluating a line integral directly
$F(x,y) = \dfrac{1}{x^2+ y^2}\langle -y,x\rangle$, and let $R$ be a circle of radius $a$, centered at the origin.
a) Why can't Green's theorem be used to evaluate $\int_R F \cdot ds$?
b) ...
0
votes
0answers
44 views
separating a variable from integral
In the following integral, I would like to separate $\alpha$ from rest of the equation. Can we solve the following equation for $\alpha$?
$$\large{\int_{0}^{a} \int_{0}^{2\pi} ...
-1
votes
1answer
31 views
Piecewise defined integration [closed]
Let
$$f_n(x) = \begin{cases}
0 & x < -\tfrac{1}{n} \\
\tfrac{n}{2} & -\tfrac{1}{n} \leq x \leq \tfrac{1}{n} \\
0 & x>\tfrac{1}{n} \\
\end{cases},$$
$n=1,2,3,\ldots$.
Let $F(x) = ...
21
votes
2answers
212 views
$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$
I need to find a closed-form for the following integral. Please give me some ideas how to approach it:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
2
votes
4answers
70 views
$\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$
I'm having trouble understanding how to apply the $\frac{d}{dx}$when taking the anti-derivative.
$$\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$$
In class it was mentioned we'll end up taking ...
1
vote
1answer
41 views
$\lim_{R \to \infty} \int_0^R \frac{dx}{(x^2+x+2)^3}$
$$\lim_{R \to \infty} \int_0^R \frac{dx}{(x^2+x+2)^3}$$
Please help me in this integral, I've tried some substitutions, but nothing work.
Thanks in advance!
-1
votes
1answer
67 views
$ \underset{r \rightarrow 0+}{\lim} \int ^1 _r x^{2013}(\ln x)^{1001} dx$
This integral is in my book, but I don't know how to solve it using simply methods. It's possible to do it clever.
$$ \underset{r \rightarrow 0+}{\lim} \int ^1 _r x^{2013}(\ln x)^{1001} dx$$
Could ...
1
vote
2answers
47 views
Evaluating the integral: $\lim_{R \to \infty} \int_0^R \frac{dx}{x^2+x+2}$
Please help me in this integral:
$$\lim_{R \to \infty} \int_0^R \frac{dx}{x^2+x+2}$$
I've tried as usually, but it seems tricky. Do You have an idea?
Thanks in advance!
27
votes
3answers
373 views
Conjectural closed-form representations of sums, products or integrals
What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
1
vote
2answers
48 views
Computing $\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$ using substitution
Consider this integral:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$$
How would you compute it?
I already solved this problem this way:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz = \left( ...
3
votes
2answers
84 views
$\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$ , $\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$
Please help me integrate
$$\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$$
and
$$\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$$
I've tried the standard $u = \tan \frac{x}{2}$ substitution but it looks ...
4
votes
3answers
91 views
Integrating left to right versus right to left.
OK, I understand that when integration is done left to right with respect to x increasing left to right (dx is positive), that the answer is positive, and vice versa when integrating right to left. ...
2
votes
1answer
33 views
Evaluating Complex Line Integrals
Calculate $\int_{\gamma}\frac{\Re(z)}{z-\frac{1}{2}}dz$ and $\int_{\gamma}\frac{\Im(z)}{z-\frac{1}{2}}dz$ when $\gamma$: $|z|=1$ is positively oriented.
This is what I have tried to do, starting ...
1
vote
0answers
47 views
$\int \frac{e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)}$
I'm stuck on my last exercise. Could you help?
$$\int \frac{e^x+1}{(e^x\sin x+\cos x)(e^x\cos x-\sin x)} \ dx$$
3
votes
0answers
31 views
What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?
I want to compute the following integral:
$\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$
with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$.
...
1
vote
3answers
43 views
Integration of a rational function from +/- infinity
I am trying to calculate the integral
$$\int_{-\infty}^{\infty}{\frac{a+x}{b^2 + (a+x)^2}\frac{1}{1+c(a-x)^2}}dx$$
where $\{a, b, c\}\in \mathbb{R}$. I have looked in a table of integrals for ...
0
votes
1answer
45 views
$ \int_{0}^{\infty}{\dfrac{\cos(ax)}{(x^2 + 1)^2}dx} $
I have a contour integral problem I need to solve, but I don't know the answer, so I wanted to verify that my work is correct.
$$ \int_{0}^{\infty}{\frac{\cos(ax)}{(x^2 + 1)^2}dx} $$
For this one, ...
2
votes
1answer
49 views
How to place a limit that it's inside the integral, outside.
I did this:
$$\int_{1}^t x^{-1}dx=\int_{1}^t\lim_{n\rightarrow -1}{x^n}dx =\lim_{n\rightarrow -1}\int_{1}^t{x^n}dx $$ just to have a way to approximate $\ln t$. $$\ln{t}=\lim_{h\rightarrow ...
2
votes
0answers
50 views
Double Integral Homework Problem
Here's the problem statement of the question which I am stuck on:
Let $R_{1}$ denote the rectangle $[0, 5] \times [-4, 4]$, $R_{2}$ the rectangle $[0, 5] \times [0, 4]$, and $R_{3}$ the rectangle ...
2
votes
2answers
31 views
Quadrature formula
How can we find a quadrature formula $\int_{-1}^1 f(x) dx=c \displaystyle \sum_{i=0}^{2}f(x_i)$ that is exact for all quadratic polynomials?
Thanks for help.
0
votes
2answers
77 views
Integral question help me?
I am in the middle of solving a diff equation and I have to solve $\int\dfrac{1}{\cos x}e^{\tan x}\,dx$? I am thinking about integration by parts but it is a very long way.is there another short way? ...
0
votes
1answer
23 views
Integral, set and parametric representation
I am to compute the following: $\displaystyle\iiint\limits_V 1\, dx\, dy\, dz$,
where $V= \{{(x,y,z) \in \mathbb R^3 : (x-z)^2 +4y^2 < (1-z)^2} \text{ and } 0<z<1\}.$
Does anyone have idea ...
2
votes
3answers
42 views
Integrating a sine function that is to an odd power
I've started the chapter in my book where we begin to integrate trig functions, so bear in mind I've only got started and that I do not have a handle on more advanced techniques.
$\eqalign{
& ...
3
votes
4answers
77 views
Computing $\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}$?
How would you compute$$\int_0^{\pi\over2} \frac{dx}{1+\sin^2(x)}\, \, ?$$
1
vote
4answers
59 views
Integral of $\int \frac{x^4+2x+4}{x^4-1}dx$ [duplicate]
I am trying to solve this integral and I need your suggestions.
$$\int \frac{x^4+2x+4}{x^4-1}dx$$
Thanks
1
vote
3answers
52 views
How to evaluate $\int_1^\infty \frac{1}{x}-\frac{1}{x+1}~dx$
It's a very simple question but it confuses me. How do I evaluate
$$
\int_1^\infty \frac{1}{x}-\frac{1}{x+1}~dx
$$
without splitting? And why can't I split it?
7
votes
1answer
140 views
Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$
Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that
$$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$
Prove that
$$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$
Where should ...
0
votes
1answer
54 views
Theorem or just a change of varibles?
I have a formula in my text:
$$\int \int_{S} F \cdot n dA= \int \int_{w} F(G(u,v)) \cdot (dG_{u}\times dG_{v}) du dv$$
I am really lazy and hate remembering formulas to me this looks like a ...
6
votes
3answers
97 views
Integral of $\int^1_0 \frac{dx}{1+e^{2x}}$
I am trying to solve this integral and I need your suggestions.
I think about taking $1+e^{2x}$ and setting it as $t$, but I don't know how to continue now.
$$\int^1_0 \frac{dx}{1+e^{2x}}$$
Thanks!
2
votes
1answer
82 views
Limit of a continued fraction
Given the continued fraction:
$$f(x,N)=\left[2,3,4,...N,x\right]$$
$$f(x,N)=\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{...+\cfrac{1}{x}}}}}$$
is it possible to find an expression for the integral:
...
1
vote
2answers
39 views
Integral of $\int(4-2x)^\frac{1}{3}dx$
I solved this integral then I did $\frac{d}{dx}$ of $F(x)$ and saw that its not the same, so I did wrong in my integration process.
$$\int(4-2x)^\frac{1}{3}dx$$
What I did is $$F(x) ...
0
votes
5answers
86 views
$\int^1_0 \frac{xdx}{x^2+2x+1}$
I need some suggestion how to solve this integral.
$$\int^1_0 \frac{xdx}{x^2+2x+1}$$
I think about to do the following step :
$$\frac{1}{2}\int^1_0\frac{2x+2-2dx}{x^2+2x+1}$$$$ t=x^2+2x+1 \rightarrow ...
5
votes
2answers
58 views
Integral of fractional expression $\int^3_0 \frac{dx}{1+\sqrt{x+1}}$
I want to solve this integral and think about call $\sqrt{x+1} = t \rightarrow t^2 = x+1$
$$\int^3_0 \frac{dx}{1+\sqrt{x+1}}$$
Now the integral is : $$\int^3_0 \frac{2tdt}{1+t}$$ now I need your ...
3
votes
1answer
51 views
Calculus II, Curve length question.
Find the length of the curve
$x= \int_0^y\sqrt{\sec ^4(3 t)-1}dt, \quad 0\le y\le 9$
A bit stumped, without the 'y' in the upper limit it'd make a lot more sense to me.
Advice or solutions with ...
16
votes
1answer
161 views
Integrating $\int^{\infty}_0 e^{-x^2}\,dx$ using Feynman's parametrization trick
I stumbled upon this short article on last weekend, it introduces an integral trick that exploits differentiation under the integral sign. On its last page, the author, Mr. Anonymous, left several ...
1
vote
1answer
56 views
Improper integral sin(x)/x converges absolutely, conditionaly or diverges?
$$\int_1^{\infty}\frac{\sin x}{x}dx$$
$$u=\frac{1}{x}$$
$$du=-\frac{1}{x^2}dx$$
$$dv=\sin xdx$$
$$v=-\cos x$$
$$\int_1^{\infty}\frac{\sin x}{x}dx=\frac{1}{x}(-\cos x)-\int_1^{\infty}\frac{\cos ...
3
votes
2answers
67 views
Integration involving roots
$$\int\frac{dx}{(1+x^\frac{1}{4})x^\frac{1}{2}}$$
This is my work:
$$u^4=x$$
$$4u^3=dx$$
$$\int\frac{4u^3du}{(1+u)u^2}=\int\frac{4u^3du}{(1+u)u^2}=-4(1+x^\frac{1}{4})^{-1}+2(1+x^\frac{1}{4})^{-2}+C$$
...
4
votes
1answer
60 views
The geometric interpretation [duplicate]
In the course of mathematical analysis, there was one problem that i excited to know more about it:
What is the geometric interpretation of
$$ \int_a^b f(x)\,d(\alpha(x)) $$
and $\alpha(x)$ is ...
15
votes
1answer
160 views
Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$
Please help me to find a closed form for the following integral:
$$\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx.$$
I was told it could be calculated in a closed form.
2
votes
0answers
64 views
A photon in expanding Universe (a snail on a tree)
I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
0
votes
1answer
24 views
How to solve expectation / first moment of Gaussian Integral?
How can I solve the following integral?
$E[I] = \int_{I=0}^\infty I \frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{(\mu-f-I)^2}{2\sigma^2}) dI$
The result is supposed to be:
$\sigma[\frac{\mu - ...
1
vote
2answers
35 views
Proving that $\frac{\sigma_{n-1}}{\omega_n} = n$ in $\mathbb{R}^n$
If $\sigma_{n-1}$ was the surface area of the unit sphere in $\mathbb{R}^n$ and $w_{n}$ was the area of the unit ball in $\mathbb{R}^n$, my lecture notes prove that $$\frac{\sigma_{n-1}}{\omega_n} = ...
3
votes
2answers
72 views
Integral of $ \int_{-1}^{1} \frac{x^4}{x^2+1}\,dx $
Any suggestions how to solve it? by parts?
$$ \int_{-1}^{1} \frac{x^4}{x^2+1}dx$$
Thanks!
