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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4
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2answers
88 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
43 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
85 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
30 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
36 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
47 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
38 views

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

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 ...
0
votes
2answers
43 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
54 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
1answer
50 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 ...
0
votes
0answers
24 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
37 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
60 views

Integral of $xe^{-ax^2-bx^{-1}}$

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
30 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
58 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
41 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
43 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
65 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
410 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 ...
2
votes
0answers
29 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, ...
1
vote
1answer
102 views

Finding the surface area of the spheroid $\frac{x^2}{3} + \frac{y^2}{3} + \frac{z^2}{4} = 1$

I'm asked to evaluate this: What is the surface area of the surface defined by $\frac{x^2}{3} + \frac{y^2}{3} + \frac{z^2}{4} = 1$? I first parameterized it with spherical coordinates and then I ...
1
vote
1answer
38 views

Surface area of ellipsoid given by rotating $\frac{x^2}{2}+y^2=1$ around the x-axis

Calculate the surface area of the ellipsoid that is given by rotating $\frac{x^2}{2}+y^2=1$ around the x-axis. My idea is that if $f(x)=\sqrt{1-\frac{x^2}{2}}$ rotates around the x-axis we will end ...
0
votes
0answers
32 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 $\delta(a)>0$ ...
3
votes
2answers
52 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 ...
0
votes
1answer
13 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 ...
0
votes
1answer
58 views

A question about improper integral

Would you please give me a hint on how to solve this problem: Suppose $f(x)$ continuous in $[0,\infty)$ and for each $a,b>0$ and $c>b$, we have \begin{equation*} ab \left|\int_0^1 f\left(ax+c ...
0
votes
1answer
37 views

How can I prove that these integrals do not converge?

Can you please help me to prove the integrals \begin{equation*} \int_0^\pi \frac{x}{\sin(x)}~\text{and}~\int_0^\infty \frac{1}{\sqrt{x}}\cos(x^{-1}) \end{equation*} are divergent? Please I really need ...
0
votes
1answer
57 views

$\lim_{y \rightarrow^{nt}x}\int \omega(y-z)g(z) \,d\sigma(z)=\lim_{\epsilon \rightarrow 0}\int_{|x-z|>\epsilon}\omega_j(x-z)g(z)\,d\sigma(z)$

We need to show $\lim_{y \rightarrow^{nt}x}\int \omega_j(y-z)g(z) \,d\sigma(z)=\lim_{\epsilon \rightarrow 0}\int_{|x-z|>\epsilon}\omega_j(x-z)g(z)\,d\sigma(z)$ Here is the necessary information ...
1
vote
1answer
129 views
+50

Functional Analysis, a question that needs clarification.

Find the norm of the linear operator $A:C[-1,1]\to L^p[-1,1]; p\geq1$ that is defined as: $$A(x(t))=\int_{-1}^{1}{{x(s)\over (s-t)^{1 \over 3}}}ds$$ Can someone provide an answer with a little more ...
0
votes
1answer
27 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 $\dfrac{5\pi}{6}$. ...
2
votes
1answer
415 views

Approximating integrals with step functions

For $f \colon [1,2] \to \mathbb{R}$ , $f(x) = 1/x$, Choose a sequence of step functions $\phi_n$ approximating $f$ with partition $P_n := [r/n : n < r < 2n]$ to show that $ 1/(n+1) + \cdots + ...
0
votes
2answers
58 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
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 $0 \leq y \leq 1$
1
vote
1answer
448 views

Change of variable (translation) in complex integral

If I have a real integral, e.g. $\int f(x+2) \ dx$, I can substitute $y = x+2$, so $dy = dx$. But if my function is complex, am I still allowed to do this? In which cases I cannot apply a ...
1
vote
0answers
14 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 ...
2
votes
1answer
77 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, ...
4
votes
1answer
47 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
votes
2answers
33 views

question about measure zero and discontinuities of a function

Let $E \subseteq \mathbb{R}^n$ be a closed set, and say $E$ has measure zero. Let $g: Q \to \mathbb{R}$ be a bounded function where $Q$ is a box in $\mathbb{R}^n$. $E \subseteq Q $. Define $$ A = \{ ...
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 ...
0
votes
0answers
17 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
votes
2answers
59 views

Find the value of the integral $\int_{-\infty}^{\infty} \frac{1}{1+(a-y)^2} \frac{1}{1+y^2} dy$

Can someone help me with the evaluation of the integral \begin{equation*} \int_{-\infty}^{\infty} \frac{1}{1+(a-y)^2} \frac{1}{1+y^2} dy? \end{equation*}
23
votes
2answers
774 views
+500

A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

I need to find a value of this definite integral: $$\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx.$$ Its numeric value is approximately $0.7875720991394284$, and lookups ...
0
votes
0answers
24 views

Stieltjes integral with discontinuous integrator

I am asked to solve the following Stieltjes integral: Compute $\int_0^6 f\, dg$, where $f(x) = 6x-x^2$ and $g(x)$ is defined by: $$ g(x) = \left\{ \begin{array}{ll} x^2 &\hbox{for $0\leq x ...