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 ...

learn more… | top users | synonyms (3)

9
votes
4answers
205 views

Solve trigonometric integral $\int_{-\pi/2}^{\pi/2} \frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x} dx $

Please help me to solve the following integral: $$\int_{-\pi/2}^{\pi/2} \frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x} dx$$ I have tried a lot, but no results. I only transformed this integral to the ...
5
votes
0answers
17 views

Evaluating $\int_0^1 \frac{z \log ^2\left(\sqrt{z^2+1}-1\right)}{\sqrt{1-z^2}} \, dz$

What real analysis tools would you employ for this kind of integral? $$\int_0^1 \frac{z \log ^2\left(\sqrt{1+z^2}-1\right)}{\sqrt{1-z^2}} \, dz$$
0
votes
1answer
21 views

Help understanding definition of Darboux integral $U(f)$.

My book defines the upper and lower Darboux sums $U(f,P)$ and $L(f,P)$ respectively then follows up with a confusing definition of the upper and lower Darboux integrals $U(f)$ and $L(f)$ respectively. ...
1
vote
0answers
19 views

Parametrization of surfaces for vector integration

I'm having some trouble calculating vector fields through surfaces. After attempting a few and being dissapointed with a wrong answer multiple times I figured I must be doing something wrong in the ...
2
votes
1answer
41 views

Function defined by integral

this question is driving me nuts, I can't think about an easy solution. Let $F(x)=\int_{0}^{x} \sqrt{1+t^3}\,dt. $ Evaluate $\int_{0}^{2} x\,F(x)\,dx$ in terms of $F(2)$. I know that the derivative ...
1
vote
1answer
37 views

Im having trouble figuring this integral out can someone help? Not allowed to use polar coordinates.

$$c\int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\sqrt{1-x^2-y^2}dy \:dx=1 .$$ Find $c.$ I went with the substitution say $b=1-x^2$ in the first integral. Then I went with : $\cos t= {y \over ...
3
votes
0answers
23 views

Weak convergence of measures iff subsequence of subsequence of distribution functions converges a.e.

I'm trying to prove the first part of Proposition 8.1.8 in V.I.Bogachev, Measure Theory 2: A sequence of signed measures $\mu_n$ on the interval $[a,b]$ converges weakly to a measure $\mu$ ...
-3
votes
1answer
57 views

Integration $e^{-x^2}$ [on hold]

How do I find the integral of $e^{-x^2}$ and $xe^{-x^2}$? And also (using these) the integral of $e^{-x^2}(x^{2n+1})$ (by integration by parts)?
5
votes
4answers
4k views

How to measure the volume of rock?

I have a object which is similar to the shape of irregular rock like this I would like to find the volume of this. How to do it? If I have to find the volume, what are the things I would need. eg., ...
-1
votes
0answers
14 views

Solving a system of Volterra integral equations

I'm studying the reliability of a mechanical system. I have a system of $n$ Volterra integral equations of the second kind with $n$ unknown functions. How am I supposed to solve it?
2
votes
3answers
38 views

On an Integral inequality.

I am following a proof and I am having troubles with the last inequality stated Specifically could I have some extra passages on this? $$\int_{\delta}^{\pi} [f(w+u) - f(w)] \frac{\sin^2(nu/2)}{2 ...
2
votes
2answers
59 views

Integral of $\frac{\sin^2(nx/2)}{\sin^2(x/2)}$ over $[-\pi,\pi]$.

I would like to show that $$\frac{1}{n\pi}\int_{-\pi}^\pi \frac{\sin^2(nx/2)}{2\sin^2(x/2)} dx = 1$$ My attempt is very similar to the accepted answer to this question. $$\int_{-\pi}^\pi ...
1
vote
2answers
17 views

Show a function satisfies the diffusion equation

Show $u(x,t) = \int_0^{x/t^{1/2}} e^{-0.25b^2}db$ satisfies $\dfrac{\partial u}{\partial t} = \dfrac{\partial ^2 u}{\partial x^2}$ How do I go about doing this? Particularly because $e^{-x^2}$ ...
2
votes
2answers
56 views

Convergence of $\int\frac{\arctan x}{x} dx$

I can't find function to bound this integral in the intervals from $1$ to $+\infty$, to prove if it converges. $$ \int _1^{\infty }\frac{\arctan x}{x}dx $$ Any idea? How can I refute this if it is ...
30
votes
6answers
3k views

Is it possible to write a sum as an integral to solve it?

I was wondering, for example, Can: $$ \sum_{n=1}^{\infty} \frac{1}{(3n-1)(3n+2)}$$ Be written as an Integral? To solve it. I am NOT talking about a method for using tricks with integrals. But ...
0
votes
1answer
13 views

Question about double summation notation.

Just started learning about double integrals literally $10$ minutes ago. I have a fairly good grip on the Riemann integral and so far it seems very similar, but we are just working with volumes ...
0
votes
1answer
42 views

Convergent of integral of $1/x^x$

I need to prove if this integral converges (on interval $[1,+\infty))$: $$\int _1^{\infty }\frac{1}{x^x}\;dx$$ Has anybody any idea how to do it? thank you. :)
2
votes
1answer
28 views

Showing an integral is in $L^1$

Let $0<a<1$ and $f\in L^1([0,1])$. Show $g(x)=\int_0 ^x\frac{1}{(x-t)^a}f(t)dt$ exists a.e. in $[0,1]$ and $g\in L^1([0,1])$. Using Fubini, $$\int_0 ^1 \vert g(x) \vert dx=\int_0 ^1 \int_0 ...
1
vote
1answer
21 views

Finding volume of a cone using triple integral

The cone has the formula: $x^2 + y^2 = z^2 , 0≤z≤2$ So I used the cylindrical coordinates to get the following answer: $$\int_0^{2\pi}\int_0^2\int_0^2 dz\,rdr\,d\theta = 8\pi$$ In the solution of ...
1
vote
0answers
26 views

Non trivial integral with the Bose-Einstein distribution and Cosine function

Do you have any idea how to solve this integral? $$\int\limits_0^\infty {\frac{{\cos \left( mx \right)}}{{x + {x_0}}}\left( {1 + n\left( x \right)} \right)} - \int\limits_0^\infty {\frac{{\cos ...
1
vote
1answer
10 views

Squared Hellinger Distance subadditive for Product measures

How can I show that the squared Hellinger Distance is subadditive for Product measures? We have $\mathbb{P} = \otimes_{i=1}^n \mathbb{P_i}$ and $\mathbb{Q} = \otimes_{i=1}^n \mathbb{Q_i}$ ...
0
votes
0answers
16 views

Explicit formula using divergence theorem

Let $A=(0,1)^k$ be the open unit cube in $\Bbb{R}^k$ and let $f\in C^1(A,\Bbb{R}^k)$. If $n$ is a unit surface normal, then by the divergence theorem, $$\int_A \text{div}f(x) dx = \int_{\partial A} ...
1
vote
1answer
44 views

Computing $\int _C \frac {1}{z^3(z-1)^2}$, $C: |z-2|=5$

How do I compute $\int _C \frac {1}{z^3(z-1)^2}$, $C: |z-2|=5$? I can't seem to use Cauchy's Formula, because both $0$ and $1$ are in the formula. There is this theorem, saying that $\int _C= ...
3
votes
2answers
65 views

How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$

I need help to prove this in real analysis. I think it uses IMVT, but not sure how to do it. Let $f(x)$ be a real valued continuous function on $[0,1]$. Show that $$ \lim ...
2
votes
0answers
19 views

If $f$ and $\alpha$ are discontinuous at a common point, then $f$ can't be R-S integrable.

We know that $f$ will be R-S integrable with respect to $\alpha$ (which belong to $BV(I)$, $I=[a, b]$) iff the set of all discontinuities forms an $\alpha$-zero set. In that proof $D:=\{x\ |\ x ...
0
votes
2answers
57 views

One point following another moving in a straight line?

There is a plane with two points on it, let's say A and B. A starts at an arbitrary constant point, let's say $(0, 0)$, and $B$ at a point that needs to be tested, which we'll call $(c, d)$. A moves ...
5
votes
2answers
127 views

Prove or disprove $\int_{-\infty}^\infty \frac{dx}{\cos x+\cosh x}=\frac{1512835691 \pi}{1983703776}$

In this question, Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$ , robjohn evaluates the integral to a nice summation with an approximate value. When plugged into W|A, it ...
1
vote
1answer
44 views

Evaluate limits:$\lim\limits_{n\to\infty}\int_0^nf_n^2\ dm$, $\lim\limits_{n\to\infty}\int_0^nf_n\ dm$ with $f_n(x)=\frac{e^{\sin(x^2/n)}}{1+x}$

So I am working through some practice problems, and on one of them I can't get the second part: For $x\in(0,\infty)$ and $n\in\{1,2,3,\dots\},$ let $f_n(x)=\frac{e^{\sin\left({x^2/n}\right)}}{1+x}.$ ...
0
votes
2answers
88 views

Suppose $a<b<c<d$ and $p(x)=(x-a)(x-b)(x-c)(x-d)$. Show that $\int_a^b \frac{dx}{\sqrt{|p(x)|}} = \int_c^d \frac{dx}{\sqrt{|p(x)|}}$

Suppose $a<b<c<d$ and $p(x)=(x-a)(x-b)(x-c)(x-d)$. Show that $$\int_a^b \frac{dx}{\sqrt{|p(x)|}} = \int_c^d \frac{dx}{\sqrt{|p(x)|}}.$$ My attempt: I perform linear substitution $u=x-a+c$ ...
1
vote
0answers
32 views

prove Gaussian integral using polar cordinates

The proof method is to equate expression$\mathrm{\iint_{-\infty}^\infty\,e^{-(x^2+y^2)}}$ (Cartesian)with $\mathrm{\int_0^{2\pi}\int_0^{\infty}e^{-r^2}drd\theta}$(polar) however, the answer goes ...
4
votes
1answer
41 views

Complex analysis integration with logs

$$\int_C \operatorname{Log}\left(1-\frac 1 z \right)\,dz$$ where $C$ is the circle $|z|=2$ I don't even know how you would begin doing this. I understand $\operatorname{Log}(z)=\ln|z|+i\arg(z)$, ...
0
votes
0answers
53 views
+50

Inverting a complex function

I am facing the following problem. I know that the following relationship holds $$A(\omega) = C + \int_{0}^{\infty} \frac{L(\tau)}{1 + i\omega \tau}\mathrm{d}\tau$$ where $C$ is a positive constant ...
0
votes
1answer
10 views

calculating center of mass of the semicircle which the density at any point is proportional the distance from the center

Assuming the radius is r, and the origin is put on the center of the semicircle. Using polar coordinates. first, because symmetry, the $\bar{x}$ is 0, now trying to find $\bar{y}$: the mass of the ...
0
votes
1answer
40 views

Show by substitution that: $\int^{xy}_{x}\frac{dt}{t}=\int^{y}_{1}\frac{dt}{t}$. [on hold]

probably it is an easy one, but I can't get my head around it. Show by substitution that: $$\int^{xy}_{x}\frac{dt}{t}=\int^{y}_{1}\frac{dt}{t}$$ Any help would be greatly appreciated.
1
vote
1answer
10 views

Function of Jointly Distributed and Convolution

Looking into the continuous case of the sum of jointly distributed RVs in an example in my textbook and there are a few steps missing that I can't seem to wrap my head around. If $X$ and $Y$ are ...
5
votes
2answers
57 views

Integral depending on a parameter

Task: find all values of the parameter, such that integral converges. $$\int_0^{+\infty} \frac{dx}{1+x^a \sin^2x}$$ I tried a lot and i used Cauchy and Weierstrass method but it was useless. And now i ...
1
vote
0answers
31 views

Divergence theorem for a second order tensor

I want to integrate by part the following integral in cylindrical coordinates $$\int \vec{r} \times (\nabla \cdot \overline{T}) ~d^3\vec{r} $$ where $\overline{T}$ is a second order symmetric tensor ...
1
vote
1answer
38 views

Different results on doing $\frac{\partial}{\partial y}\left(\int_r^y \frac{1}{\sqrt{y^2-s^2}} ds \right)$ in different ways

I have a confusion when trying to get the result of the expression below, $$ I = \frac{\partial}{\partial y}\left(\int_r^y \frac{1}{\sqrt{y^2-s^2}} ds \right). $$ All variables are real and $y>r$. ...
1
vote
0answers
35 views
1
vote
1answer
64 views

Integral of $xe^{-ax^2-bx^{-1}}$ [on hold]

I am currently facing an integral I have no clue how to solve it. I believe it is rather exoctic, but I hope you might have some good advice: $$\int_0^{\infty} x e^{-ax^2-bx^{-1}} \, \mathrm{d}x, ...
1
vote
0answers
33 views

Calculate $\displaystyle\int_{-T}^T\sin(x-a)\cdot\sin(x-b)~e^{-k~(x-a)(x-b)}~dx\quad$

Calculate $$\displaystyle\int_{-T}^T\sin(x-a)\cdot\sin(x-b)~e^{-k~(x-a)(x-b)}~dx\quad$$ I have no idea how to proceed. Any suggestions please? Here $T>0$.
0
votes
0answers
21 views

Given a set of points, find the plane parallel to plane $p$ where your plane cuts the area in half.

Given a set of point $G=\{(x,y,z) | 0 \le x\le2, 0 \le y \le 2, 0 \le z \le xy\}$ for all $x,y>0$ Find the plane $p$ parallel to plane $zy$ whereas you get two areas equal in size What I did was ...
0
votes
1answer
59 views

Exponential integral of sine

How can I calculate the following integral: $$ \int_{-\infty }^{\infty} e^{-x^{2} + sin x}dx$$ Thank you very much!
1
vote
2answers
43 views

under which conditions this equality holds

Consider $f : [0,\infty) \rightarrow \mathbb{R}$ be a function such that $\lim_{t\rightarrow \infty} f(t) = 0$. I was wondering if the following relation holds $$lim_{t\rightarrow\infty}\int_0^t ...
2
votes
2answers
49 views

definite integral of a complex function

I wonder if there is a way to evaluate this definite integral... $$\frac{2}{\pi}(\ln (2) + \int_{0}^{\infty}({\sqrt{\frac{1}{t^{4}} - \frac{4e^{-4t}}{(1 - e^{-4t})^{2}}}} - \frac{1}{t^2 + 1})dt)$$ ...
2
votes
2answers
67 views

Expectation of $\mathbb{E}(X^{k+1})$

I have difficulties with an old exam problem : Let $X$ be a positive random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$. Show that $$\int_0^\infty t^k ...
1
vote
2answers
10 views

Discretization of integral on infinite domain.

Let $[a, b]$ be a closed interval of the real line and let a sequence as $$a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!$$ This partitions the interval $[a, ...
6
votes
1answer
176 views

Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a function of $\mathcal{C}^2(\textbf{R}^d$;$\textbf{R}^*_+)$ such that $G \in \operatorname{L}^1(\textbf{R}^d)$. I read on the Internet that one has the following equivalence ...
2
votes
3answers
413 views

Calculate the area of the ellipsoid obtained from ellipse $\frac {x^{2}}{2}+y^{2} = 1$ rotated around the $x$-axis

So we are about to calculate the area of the ellipsoid around the $x$-axis. $$ \frac {x^{2}}{2}+y^{2} = 1 \implies x=\sqrt{2-y^{2}}$$ We are squaring it so the sign shouldn't matter. I was ...