Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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1
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3answers
20 views

Riemann integral on trigonometric functions

I have to calculate Riemann integral of function $g:[0;\pi/4]\rightarrow\mathbb{R}$ (on interval $[0;\pi/4]$) given as $g(x)=\frac{\tan(x)}{(\cos(x)^2-4)}$. Function $g$ is continous on interval ...
-3
votes
1answer
40 views

Find area under the given curve [on hold]

Find the area under the curve $$\displaystyle y = \frac{|x-3| + |x+1|}{|x+3| + |x-1|},$$ the $x$-axis , $x = -3$ and $x = 1.$
5
votes
2answers
189 views

Estimating an integral

This question is a more specific question related to http://mathoverflow.net/questions/69900/asymptotics-for-the-number-of-ways-to-sum-primes-such-that-the-sum-is-n I am looking for a lower bound ...
3
votes
1answer
35 views

How to find the integral $\int_0^zexp(ax)x^{b-1}(1-x)^{c-1}dx$?

How to find the integral $\int_0^z exp(ax)x^{b-1}(1-x)^{c-1}dx,~b,c\in \mathbb{C}, Re(b)>0, Re(c)>0$?
5
votes
1answer
34 views

green's second identity application

I need to use the green's second identity in order to prove the following equality: $$ \int_{\mathbb{R}^2} \ln (\sqrt{x^2+y^2})\Delta f = -2\pi f(0)$$ where $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ ...
2
votes
3answers
114 views

Finding $\int ^\pi_{-\pi}\sin(nx)\cos(mx)dx$

Find $$\int ^\pi_{-\pi}\sin(nx)\cos(mx)dx$$ I used the product identity and got: $\displaystyle \int ^\pi_{-\pi}\sin(nx)\cos(mx)dx = ...
6
votes
3answers
121 views

How do I compute this integral?

I'm wondering how to compute the integral $$ \int_2^3\int_0^\sqrt{3x-x^2}\frac{1}{(x^2+y^2)^{1/2}}\,\mathrm{d}y\mathrm{d}x. $$ Clearly it is too complicated to do it directly, so I'm guessing you have ...
0
votes
1answer
18 views

Upper bounding incomplete gamma function

For $0<\theta, \lambda<1$ and $c>1$, we wish to upper bound the following gamma function: $$\int_{\theta}^{1} t\exp \left(-c\left(\lambda t+\frac{1}{t}\right) \right)dt$$
37
votes
4answers
2k views

Proof of Frullani's theorem

How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously ...
3
votes
4answers
107 views

How to solve $\int e^{-\sqrt{x}}dx$

I have this integral: $$\int e^{-\sqrt{x}}dx.$$ This is what I have done: $$\int e^{-\sqrt{x}}dx = \int \frac{1}{e^{\sqrt{x}}} dx$$ I Tried to solve it by substitution: $$t = \sqrt{x}$$ $$ t^2 = ...
2
votes
0answers
10 views

Question about an integral involving a miminum in the integrand

Say I have an integral of this form: $\int_{0}^{min(x,y)}f(x,u)f(y,u)du$ and I want to get it in this form: $\int_{0}^{y}f(y,u)du\int_{0}^{x}f(x,u)du$, does anyone know of any practical way to do ...
0
votes
1answer
23 views

Integration with vectors

I am trying to solve below integration $$\int_{0}^{\infty}\hat{k}\frac{e^{ikR}}{k-l}dk$$ here $R,l$ are constants and $\hat{k}$ is a unit vector of $\textbf{k}$. And as usual ...
0
votes
0answers
23 views

Connection between Dirichlet series and integration?

For quiet sometime I've been working on an idea of mine (I think its not discovered at least, within my googling capabilities). I think I found a connection between the Dirichlet series and ...
0
votes
2answers
22 views

Evaluate convolution integral

Can someone tell me if I am calculating this integral correctly.
4
votes
3answers
198 views

Problem of Integration by Parts involving algebraic and exponential functions

Can anyone please help me in solving this integration problem $\int \frac{e^x}{1+ x^2}dx \, $? Actually, I am getting stuck at one point while solving this problem via integration by parts.
0
votes
1answer
11 views

Convolution integral involving two Heaviside functions

I am having trouble solving the following integral involving two Heaviside functions, obtained from a Laplace transform convolution: $\Large \int_0^t \frac{\tau}{\sqrt{\tau^2 - \alpha^2}} H(\tau - ...
1
vote
1answer
26 views

Calculate the flux through a closed surface

While studying for a test I have encountered such a task: Calculate the flux through a closed surface, where $S$ is a boundary of area $V$ with an outward orientation. The data: ...
0
votes
2answers
17 views

Integral differentiation with infinite bound (differentiation of expected value)

I am trying to proove the following: $$\frac{d}{dx}\int_x^{\infty}(z-x)f_Z(z)dz=1-F_Z(x)$$ Where $f_Z$ and $F_Z$ are resp. the probability density and cumulative distribution functions of a random ...
0
votes
1answer
11 views

Calculate the flux through a surface S from a field described by vectors

I have encountered yet another example which is not that typical. I need to calculate: $$\iint\limits_{S} \vec{F} \vec{ds} =\text{ ?}$$ Where the $F$ and $S$ are as follows ($S$ is oriented ...
1
vote
2answers
99 views

Can I solve this integral with a squared sum in it?

Title says it all. By now I have tried by hand and I think that it is indeed solvable, but I can't handle the very long terms. I tried to run the thing through SAGEs integrator: ...
0
votes
3answers
61 views

Integration $ \int x^2 \cos(nx) dx $

How do I solve this integral? $ \int x^2 \cdot \cos(\frac12 n\pi x)dx $ I tried integration by parts... $ u=x^2\\u'=\frac13 x^3\\v'=\cos(\frac12 \pi n x)\\v=sin(\frac12 \pi n x)\cdot \frac12 \pi n ...
6
votes
2answers
126 views

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

How could we prove that $$\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~=~a\cdot2^{a-1}~\bigg[\frac12~B\bigg(\frac a2,~\frac a2\bigg)~+~B\bigg(\dfrac{a+1}2,~\dfrac{a+1}2\bigg)\bigg],$$ where ...
7
votes
1answer
162 views

Finding fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Show that when $n=2$, the function $u(x)=-\dfrac{1}{8\pi}\left\lvert x\right\rvert^2\log\left\lvert x\right\rvert$ is a fundamental solution to the biharmonic operator ...
0
votes
2answers
66 views

How to solve this integration?

I want to solve this $$\int_0^w (b/x)^{a+1} e^{(cx-(b/x)^a)} dx$$ where $a$, $b$, and $c$ are arbitrary positive real numbers. Do i have to solve it numerically? I have no clue to solve this ...
0
votes
0answers
23 views

HJM Model vs Leibniz integral rule

I state that I'm an electronic engineer (undergraduate), then the my knowledges about advanced mathematics are almost null. A colleague asked to me an help about one point of the proof of the theorem ...
4
votes
1answer
89 views

for which values of $\alpha \in \mathbb R$ is $f$ integrable?

For which values $\alpha \in \mathbb{R}$ is $f$ integrable? $$f: \mathbb{R}^2 \rightarrow \mathbb{R} : f(x, y) = x \frac{\ln(1 + x^2 + y^2)}{(x^2 + y^2)^\alpha} $$ if $ (x,y) \neq (0, 0) $ and $ ...
0
votes
0answers
18 views

upper-band of the Integral expression

Consider below integral expression $$\int_{0}^{\infty}g(y)[\int_{a}^{\infty}(1-e^{-(k+y)x})f(x)dx ]dy \ \ \ \ (1)$$ Where, we know: $$f(x)>0\ ,\ \ a\leq x \leq \infty$$ $$\ k>0$$ $$g(y)>0\ ...
-1
votes
0answers
25 views

On the integration of a Lebague measurable function [on hold]

Sincerely need help on this question, anyone who has any ideas please don't hesitate to tell me :)
0
votes
2answers
71 views

Integral $\frac{\sqrt{1+x^2}}{x}$

I understand that I will be using trig substitution, and tangent will be what is used, but I get confused later down the road when integrating with the trig.
1
vote
1answer
25 views

Finding the root mean square of a sum of trig functions

$$v(t) = 3 - 2\sin(t) + 8\sin^2(t)$$ To find the rms of this function, I first figured out that the period $T = 2\pi$. I then set up the equation: $V = \sqrt{\frac{1}{T}\int^T_0v^2(t)\,dt}$ ...
1
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0answers
17 views

When is the riemann integral of a finite-variation process square-integrable?

Let us say $X_t$ is a continuous finite-variation process and $f(t, x)$ a $C^{1, 1}$ function. We define $$ Y_t = \int^t_0 f(s, X_s)\, \mathrm ds $$ Are there any general results pertaining to ...
2
votes
0answers
58 views

solving definite integral problems without complex line integral

It is well known that some definite integrals such as $$\int_{0}^{\infty} \frac{dx}{a+\cos{x}}$$ $$\int_{0}^{\infty} \frac{\sin{x}}{x}dx$$ are solved by using complex analysis techniques. (It uses ...
3
votes
2answers
40 views

Trouble solving this differential equation: $x'=3(x-2)$, $x(0)=-1$.

Find the solution of the differential equation x'=3(x-2) given initial value condition of x(0)=-1 Here's my attempt. x'=3(x-2) dx/dt = 3(x-2) dx/x-2 = 3dt int dx/x-2 = int 3dt+c ln|x-2| = 3 + C ...
1
vote
1answer
21 views

Divergence theorem and applying cylindrical coordinates

This time my question is based on this example Divergence theorem I wanted to change the solution proposed by Omnomnomnom to cylindrical coordinates. $$ \iiint_R \nabla \cdot F(x,y,z)\,dz\,dy\,dx = ...
0
votes
0answers
33 views

Real integral done by complex methods [duplicate]

$\int_{-\infty}^{\infty} \frac{cosx}{x^2+25} dx $ = $ \frac{\pi}{5e^5}$ Any ideas?
0
votes
2answers
32 views

Which is the justification for this indefinite integral relation? [on hold]

Why is the following indefinite integral equation correct: $$ \int \frac{\cot(x)}{\sin^2(x)} dx= -\frac{1}{2}\cot^2(x) $$ What are the necessary steps?
1
vote
0answers
20 views

Bound on the mean value of function involving Hilbert transform

Consider the integral $$\int_{-\infty}^{\infty} x|A|^2_x\mathbb{H}(|A|^2_x) \ dx,$$ where $A=A(x,t)$ is a complex valued, compact function (I mean this in the heuristic sense that $A$ vanishes ...
0
votes
1answer
23 views

Volume of revolution on an area crossing the axis

Suppose a question asks for the volume of revolution about the x axis to be found on a piece of area enclosed between 2 graphs, where the area crosses the x-axis. In this case, the method involving ...
0
votes
0answers
22 views

Switching the order of integration [duplicate]

I need to switch the order of integration for the following function: $$ \int_{0}^{1} \int_{0}^{1-x} \int_{0}^{x+y} f[x,y,z] dz dy dx $$ to the order: $$ \int_{?}^{?} \int_{?}^{?} \int_{?}^{?} ...
0
votes
0answers
27 views

About a curve and its direction at every point

Consider the following class of problems: Consider a plane (or $\mathbb{R}^n$ in general). Let to any point of the plane corresponds a set of directions (set of unit vectors, I mean). Does there ...
25
votes
3answers
653 views

Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$

Does $~\displaystyle{\LARGE\int}_0^1\frac{\text{arctanh }x}{\tan\bigg(\dfrac\pi2~x\bigg)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$ possess a closed form expression ? This recent ...
0
votes
1answer
93 views

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $?

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $ ? I am completely at a loss , please help , thanks in advance .
4
votes
4answers
56 views

I'm stuck in this one of trig substitution for fuctions.

I got this: $$\int\frac{dx}{\sqrt{(4x^2-9)^3}}.$$ I know that the answer is: $$\frac{x}{9*\sqrt{4x^2-9}}+c.$$ And with the steps that I know about this type of substitution, I came up here, but.. ...
4
votes
1answer
924 views

Lemma about the integral of a function with compact support

Lemma 16.4 (p. 140) of Munkres' Analysis on Manifolds says: Let $A$ be open in $\mathbb{R}^n$; let $f: A \rightarrow \mathbb{R}$ be continuous. If $f$ vanishes outside a compact subset $C$ of $A$, ...
7
votes
2answers
68 views

Summation of the reciprocals of the product of consecutive integers

It is well known that there is a closed formula for: $$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{(n)(n + 1)}$$ And likewise for: $$\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot ...
2
votes
8answers
113 views

Integrating $\int \frac{dx}{1+\frac{a}{x}}$

First-year problem: How do I integrate $$ \int \frac{dx}{1+\frac{a}{x}}? $$ My guess is $u$-substitution, or partial fractions, but nothing that I try seems to work...
-5
votes
0answers
20 views

What method we use when n = odd for evaluate the integral using simpson's rule ??? [on hold]

hi any one can tell me What method we use when n = odd for evaluate the integral using simpson's rule ??? plz help....
15
votes
1answer
280 views

Proving that $\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$

How can we prove, without employing the aid of residues or various transforms, that, for $n>2$ $$\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$$ Motivation: ...
0
votes
1answer
37 views

Can you explain the result in this one, please?

I tried complete the square, but it doesn't work I got this: $\int\frac{xdx}{\sqrt{3-2x-x^2}}$ And I know that the answer is: $-{\sqrt{3-2x-x^2}}-\arcsin(\frac{x+1}{2})+c$
0
votes
1answer
10 views

Solving Poisson's Equation in 1-D for a point charge?

Ok so I was trying to solve the Poisson's equation for a point charge with a Fourier transform to get the familiar equation. This is what I did so far: So ultimately I am trying to solve this in 3 ...