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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0answers
15 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 ...
0
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0answers
18 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. ...
2
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1answer
40 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 ...
7
votes
4answers
168 views

Solve trigonometric integral

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 ...
1
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1answer
36 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
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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)?
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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?
1
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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 $e^{-x^2}$ ...
1
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3answers
36 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
55 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 ...
0
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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 ...
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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. :)
1
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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
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0answers
24 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 ...
0
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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} ...
3
votes
0answers
22 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 $\mu$ ...
1
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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 $\mathbb{Q} = \otimes_{i=1}^n \mathbb{Q_i}$ ...
2
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1answer
26 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
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1answer
43 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= ...
2
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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 ...
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
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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 ...
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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
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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 ...
4
votes
1answer
40 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
2answers
56 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 ...
0
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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 ...
2
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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
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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 $T>0$.
5
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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 ...
2
votes
2answers
48 views

definite integral of a complex function

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

integral inequalities and continuous functions [on hold]

Let $f$ be a positive, continuous function on $\mathbb{R}$. Let $c\in (0,1/2)$ be a constant and $\lambda>1$. I want to prove that: (1). for any $a\in\mathbb{R}$, there exists $\delta(a)>0$ ...
0
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1answer
16 views

Find the density function of $X$, from the random vector $(X,Y)$ if the PDF of this vector is:

$$\phi(x,y)= \frac{|x|}{\sqrt{8 \pi}}e^{-|x|- \frac{1}{2}x^2y^2}, x,y \in R $$ Now I'm aware I would have to do $$\phi_X(x)=\int_{- \infty}^{+ \infty}\phi(x,y) dy$$, what is confusing me is this ...
1
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0answers
35 views

How to integrate $\int dx \frac{1}{\cosh^2 x +a^2}$ [on hold]

How to integrate: $$\int dx \frac{1}{\cosh^2 x +a^2}$$
3
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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
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0answers
30 views

Finding area of a spheroid

Let $M=\{(x,y,z)\in \Bbb{R}^3 : (x/a)^2 + (y/b)^2 + (z/c)^2 = 1\}$. Find $\text{vol}_2(M) = \int_M 1 dS$. My attempt: The map $$\Phi:(0,\pi)\times (0,2\pi)\to \Bbb{R}^3\\ \qquad (\varphi, ...
0
votes
2answers
60 views

Integral of $e^{x^3}$

How do I find the integral of $e^{x^3}$. I have to do find the following integral and when I try to do integration by parts, I cannot find the integral of $e^{x^3}$. $$\int x^2 e^{x^3} ...
0
votes
1answer
28 views

Calculating volume by disc integration

What is the volume $V$ of the object created when the area formed by the lines $$y=x$$ $$y = 2-x^2$$ $$0 \le y \le 2$$ is rotated around the $y$-axis? It says that the answer is $\dfrac{5\pi}{6}$. ...
0
votes
2answers
26 views

Surface area of revolution of curve

I am wondering why this particular integration is being found difficult to solve. Would appreciate any help I can get. the graph is $y = x^3$ and the limits are $0 \leq y \leq 1$
2
votes
1answer
78 views

$1-1+1-1+1-1+\cdots = \frac 12$ proof?

Note: I claim no credit for this "proof". A friend came up with it and I thought it was pretty cool. Let's say you wanted to prove that $$1-1+1-1+1-1+ \cdots = \frac 12$$ Well, as mentioned before, ...
1
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0answers
15 views

Convergence in distribution of distributions $p_n$ implies convergence in distribution of $s_n$?

Question Setup Suppose $p_n(x,y)$ is a sequence of probability densities on $\mathbb R^2$ and $q_n(x)$ is a sequence of densities on $\mathbb R$ such that \begin{align*} \int b(x,y) \ p_n(x,y) \ dx ...
0
votes
1answer
22 views

Measurable function growing at most linearly

Let $F$ be a measurable function on $\mathbb{R}$ which grows at most linearly ($F(x) \leq C|x|$), and is differentiable at zero, $F'(0)=a$. Show that $$\lim_{n\rightarrow \infty}\int_{-\infty}^\infty ...
4
votes
1answer
48 views

Improper Intergral (Fresnel- like)

Let $\alpha >1$. Show that $$\int_0^\infty \sin(x^\alpha)\,dx= \sin\left(\frac{\pi}{2\alpha}\right) \int_0^\infty e^{-r^\alpha}\,dr.$$ I was going to ask how to do this but figured it out while ...
0
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0answers
18 views

Understanding Verlet Velocity Method

How does the Velocity Verlet method differ from the standard Euler method? Why do we need to add Acceleration / 2 to calculate position?
1
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0answers
30 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 ...
-2
votes
1answer
46 views

How to integrate a function with a nested absolute value: $|x^2 - 2|x||$? [on hold]

I need help with this problem, $$\int_0^4|x^2 - 2|x||dx$$ what should I do with $2|x|$ ?
0
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0answers
19 views

Cylindrical Coordinates

What is the following integral in cylindrical coordinates? I did it like this and I get pi/5, but if I solve in cartesian I get 2/5.