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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0
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1answer
40 views

How would you solve this surface integral?

Suppose you had the surface integral $\iint \limits_{A} = x^{3}(1-x^{4}-y^{4})dx \ dy$ where $A$ is the region defined by $x \geq 0, \; y \geq 0, \; x^{4}+y^{4} \leq 1$. How would you solve this ...
1
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0answers
22 views

Calculate Norm Operator

I'm trying to solve this exercice: Let $\omega(y)=y^{-4}$ and $L^{1}(\mathbb{R},\omega)$ the space of measurable functions $g:\mathbb{R}\rightarrow\mathbb{R}$ so that $g\omega$ is Lebesgue ...
1
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2answers
42 views

Integration (by parts?)

I don't know how the answer is obtained in the following. $2\lambda^2 \int_{0}^{\infty}ye^{-\lambda(1+z)y}dy = \frac{2}{(1+z)^2}\int_{0}^{\infty}ue^{-u}dy = \frac{2}{(1+z)^2}, 0<z < 1$ I tried ...
3
votes
3answers
80 views

Evaluation of $ \int_{0}^{\infty}\frac{x\ln x}{(1+x^2)^2}dx$

$\bf{My\; Try:}$ Let $$\displaystyle I = \int_{0}^{\infty}\frac{x\ln x}{(1+x^2)^2}\,dx = \underbrace{\int_{0}^{1}\frac{x\ln x}{(1+x^2)^2}\,dx}_{I_{1}}+\underbrace{\int_{1}^{\infty}\frac{x\ln ...
1
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1answer
26 views

Double integral over the region enclosed by 4 curves.

I'm a bit stuck on a problem that involves trying to integrate the area between 4 curves. Problem: $\int \int_D x^2+y^2 dxdy$ Where D is the region enclosed by the curves $xy =2, xy=7, y= 2x^2$ and ...
0
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0answers
53 views

Inverse Fourier transform of power functions $\omega^\alpha$ and $i\omega^\alpha$

Are there inverse Fourier transforms for the functions $F(\omega)=\omega^\alpha$ and $F(\omega)=i\omega^\alpha$ where $0<\alpha<1$ ? It seems that integrals do not converge in these cases. ...
1
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2answers
90 views

Find the length of $\sqrt x$.

Let $f(x) = \sqrt x$. Find the length of the curve for $0\le x \le a$. So I know the formula is: $$\int_0^a \sqrt{(1+(f')^2(x)} \ dx = \ldots = \int_0^a \sqrt{1+\frac{1}{4x}} \ dx$$ Now, how do I ...
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0answers
25 views

Fubini or DCT for interchange of integration and expectation

I'm trying to justify interchanging expectation and integration in the proof for positive semi-definiteness of co-variance kernels, namely, tst: $$\int_s \int_t f(s)f(t) E\{X_tX_s\} ds dt \geq 0$$. ...
0
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0answers
16 views

Proof of convergence theorem for extended Riemann Integral.

The theorem states: Suppose that f is continuous on $[a,b[$ and that $f(b^-) = + \infty$. Say $\alpha > 0$ and $\lim_{x\to b} (b-x)^\alpha f(x) = K_\alpha, K_\alpha \in \mathbb{R}\cup \{+\infty ...
1
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2answers
80 views

Application of Green's Theorem leading to different solutions of area integral

I have an area integral problem over an irregular convex polygon. I use Green's Theorem to convert the area integral to a contour integral, and solve using standard methods. Green's Theorem says I ...
1
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0answers
21 views

Algebraic Question on Riemann integration

If $f$ is Riemann integrable on $[a,b]$, then so is mod $f$. Why converse is not true?
2
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0answers
51 views

Change of variable for integration with respect to Haar measure

I know how to estimate the integral \begin{gather} \int f(Ub)\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1] \end{gather} where $f:S^n(\mathbb{R})\to \mathbb{R}$ ...
1
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3answers
39 views

I'm having some trouble with this definite integral

Im trying to evaluate this integral: $$\int_{0}^{\Pi /3}sin(2x)e^{Cos(2x)}dx$$ I have tried using U substitution letting u = 2x, and du = 2dx, giving $$\frac{1}{2}\int_{0}^{2\Pi ...
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0answers
12 views

$\int_{\gamma(.,s)}f(z)dz$ is independet from s (homotopy lemma)

Let $U\subset\mathbb{C}$ open, $f:U\to\mathbb{C}$ continuous and complex differentiable. Let $\gamma:[a,b]\times [c,d]\to U$ be in $C^2([a,b]\times [c,d])$. And assume that one of the following ...
0
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0answers
21 views

Mapping of infinite domain

I am integrating an ODE. For simplicity let us consider a simple one: $\dot{x} = u$ The solution is simply, $x = ut + x_o$. Now suppose my $x$ goes from 0 to $\infty$. But I want to do the ...
1
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0answers
16 views

Singular Parametrizations of Curves

Say $C$ is the upper half of the unit circle $x^2 + y^2 = 1$, oriented to the right. If $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is a continuous function, and $\mathbf{r}(t)$, $a \leq t \leq b$ is a ...
5
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1answer
86 views

Evaluate the improper integral $ \int_0^1 \frac{\ln(1+x)}{x}\,dx $

I am trying to evaluate $$ \int_0^1 \frac{\ln(1+x)}{x}\,dx $$ I started by using the Taylor series for $\ln (1+x)$ $$\begin{align*} \int_0^1 \frac{\ln(1+x)}{x}\,dx &= ...
1
vote
1answer
73 views

How do I integrate $\frac{x+1}{\log x}$?

How do I evaluate $I= \int_0^1 \frac{x+1}{\log x} dx$? This seemed easy at the first glance, but I could not solve this. Is there an elementary way to evaluate this integral?
0
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1answer
36 views

How to show convergence of integration when domain is moving.

Let $Q\subset \mathbb R^2$ be a cube centered at $(0,0)$, with side length $2$. Let $I$ denote the segment from $(-0.5,0)$ to $(0.5,0)$. Define $$\tau(x):=\operatorname{dist}(x,I)$$ for $x\in Q$. ...
0
votes
1answer
38 views

How to do these two integral questions? [closed]

First question: I try to use partial fraction to separate the rational function ( I don't know if I am in the right direction). However, I am still stuck. Second quesiton: I need to find the area ...
1
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2answers
44 views

The convolution of an integrable function with a $p$-integrable function is integrable

Let $\Sigma$ denote the set of Lebesgue-measurable subsets of $\mathbb{R}$, and $m$ the Lebesgue measure on $\mathbb{R}$. Let $1<p\leq \infty$, $f\in L^1(\mathbb{R},\Sigma,m)$, and $g\in ...
-1
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1answer
41 views

Find the volume above the $x-y$ plane, under the surface $r^2=2z$, and inside $r=2$

Find the volume above the $x-y$ plane, under the surface $r^2=2z$, and inside $r=2$ I'm taking this calc 3 class online and I'm completely lost this week with double and triple integrals. I've ...
1
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1answer
73 views

Evaluating $\int_0^1 x \, \text{d}(x^2)$ using Sums

How would one integrate $\int_0^1 x \, \text{d}x^2$ using the definition of the integral (no substitutions or FTC)? Is this considered a Riemann integral, and what's the physical interpretation of ...
9
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2answers
44 views

$\int_0^{f(a)} f^{-1}(t)\,dt + \int_0^a f(t)\,dt = af(a)?$ [duplicate]

If $f: [0, a] \to \mathbb{R}$ is continuous and $1$-to-$1$ with $f(0) = 0$, how do see that $$\int_0^{f(a)} f^{-1}(t)\,dt + \int_0^a f(t)\,dt = af(a)?$$
0
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0answers
44 views

Integral Solution

During my attempt to solve the non-linear ODE \begin{equation} m\ddot{x}+x-x^3=0 \end{equation} I have stumbled across the integral: \begin{equation} \int{\frac{1}{\sqrt{\frac{1}{m}\left( ...
4
votes
2answers
89 views

Prove Piecewise Function Integrable

$$ f(x) = \begin{cases} -2, & \text{if }x < 0 \\ 1, & \text{if }x > 0\\ 0, & \text{if }x = 0 \end{cases} $$ Hey guys I need some help showing that this function is integrable on the ...
1
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0answers
23 views

Loop integral with Kronecker-delta notation

I worked through a problem and arrived at the final solution (which is correct), however, one part of it should equal zero mathematically. This is the part that should equal zero: $F_{2i}= ...
1
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2answers
45 views

Show that $\int_0^\infty \left|\frac{\cos x}{1+x}\right| \ dx$ diverges

Show that $$\int_0^\infty \left|\frac{\cos x}{1+x}\right| \ dx$$ diverges My Try: $$\int_0^\infty \left|\frac{\cos x}{1+x}\right| \ dx \ge \sum_{n=0}^\infty \int_{\pi n-\pi/2}^{\pi n + \pi/2} ...
3
votes
2answers
49 views

Asymptotic complexity of sum of poly-logarithmic functions

I'm trying to figure out what's the asymptotic complexity for the following sums: $$\sum_{k=1}^n lg^s k$$ $$\sum_{k=1}^n k^rlg^s k$$ s and r are positive constants. I think i should be using ...
4
votes
2answers
62 views

Show equality of two integrals: $\int_0^\infty\frac{\cos x}{1+x}dx = \int_0^\infty\frac{\sin x}{(1+x)^2}dx$

I need to prove that: $$\int_0^\infty \frac{\cos x}{1+x} \ dx = \int_0^\infty \frac{\sin x}{(1+x)^2}\ dx$$ but, one converges absolutely whereas the other is not. I've tried few things like ...
1
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0answers
28 views

Is $\int \frac{\ln(1+\exp(a_1x-a_2))}{1+\exp(a_3x^2+a_4x+a_5)} dx$ analytically integrable?

The integral $\int\limits_0^\infty \frac{\ln(1+\exp(a_1x-a_2))}{1+\exp(a_3x^2+a_4x+a_5)} dx$ seems like it is not analytically integrable. Nevertheless, how could I construct an analytical ...
1
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0answers
28 views

Question about the Dominated convergence theorem

Suppose we have a sequence of continuous functions $f_n(x)$ for which $\int_0^\infty |f_n(x)|dx$ (Riemann) exists with $|f_n(x)|\le g(x)$ (g(x) L-integrable) for all n and $\lim_{n\to ...
0
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0answers
22 views

Recursive formula for an integral involving multiple inner products

Motivation: I am trying to form a Bayesian model where I will be performing frequent state-updates. I am seeking to find a recursive formula for a certain quantity that will enable me to perform this ...
2
votes
2answers
30 views

Computing an integral from a limit: $\lim_{n\to\infty}\sum_{k=1}^n\frac{n+k}{n^2}$

I want to express the following limit $$\lim_{n\to\infty}\sum_{k=1}^n\frac{n+k}{n^2}$$ as an integral. And I'm not sure how to compute this integral by interpreting it as the area of a known ...
0
votes
1answer
32 views

Show that $f^T$ is measurable

Suppose that $F \in~L^{+}$. Show that $$ F^T = \begin{cases} 1 & \text{if }f(x)\le 1, \\[3pt] f(x) & \text{if }1<f(x) <2, \\[3pt] 2 & \text{if }f(x)\ge 2, \end{cases} $$ is ...
0
votes
3answers
72 views

Derivative of $\ln|-\cos(x)|$

I had a Calculus 1 test on inverse functions today and one of the questions asked "What is the antiderivative of $\tan$?". I know now that the right answer is $\ln(\sec x) + C$, but the answer I put ...
0
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0answers
57 views

$\int_{A}f$ exists then $\int_{A}|f|$ exists and then $\int_{A}|f|$ > |$\int_{A}f$|

I have a real valued function $f$ on a subset $A$ of $E^n$. $\int_{A}f$ exists. Edit: I am looking at trying to show $\int_{A}|f|$ exists and then I want to show that $\int_{A}|f|$ >= ...
1
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1answer
50 views

Show that $F(x) = \int_{-\infty ,x}f$ is continuous on $\mathbb{R}$

Let $f \in L^1(m)$ where $m$ is the standard Lebesgue measure. Show that $$F(x) = \int_{-\infty}^{x}f \text{ is continuous on } \mathbb{R}.$$ I think what is confusing me is that my professor labeled ...
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0answers
31 views

Finding $\mathrm{vol}(f^{-1}(y))$

Consider a many-to-one function $f:X\to Y$ $X$ is a high-dimensional vector space and $Y$ a low-dimensional vector space The analytical form of $f$ is known You are probably free to make any 'nice' ...
2
votes
1answer
40 views

inverse laplace transform of $\frac{s^3}{(s^2+4)^2}$

Using partial fractions gives $\frac{s}{s^2+4}$ - $\frac{4s}{(s^2+4)^2}$ Inverse laplace transform of the first member ($\frac{s}{s^2+4}$) is cos(2t). Can't figure out how to transform ...
1
vote
1answer
38 views

Differential Equation problem

Find $y$, when $xy'-x=1$ and $y(1)=2$. I get that $y=x+\ln{ x}+C$ But then $y(1)=2$ isn't true. Did I do something wrong?
0
votes
1answer
47 views

How to get the equality?

How to get the equality 1,2,3 in picture below .$M$ is compact Riemannian manifold, i,e, $\partial\Omega=\varnothing$. In fact ,I have a relaxed compute about there 3 equality.I just want see the ...
7
votes
2answers
337 views

evaluating using fundamental theorem of calculus

Let $$F(x)=\int_0^{x}\ tf (x^2-t^2)\,dt$$ Find $F'(x)$. I know that I need to apply the fundamental theorem of calculus. As for the next part, I tried to substitute in $u=x^2-t^2$ but I don't ...
2
votes
3answers
42 views

Line integral along a parametrised path proof

I need some help starting the following proof, where $\vec F=x(\hat \imath+\hat k)+2y\hat \jmath$. Prove that for all paths $\Gamma$ running from $\hat \imath$ to $\hat \jmath$ and lying in the ...
1
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2answers
27 views

Calculate marginal densities

Let $f(u,v)=\mathbb{1}_{\{0\le v \le 2u\}} \cdot \mathbb{1}_{\{0\le u \le 1\}}$ How can I calculate marginal densities? I know $f(u)=\int_{-\infty}^{\infty} f(u,v)dv$ and ...
2
votes
1answer
35 views

Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?

According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that : $$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$ where: $X$ is separable real Banach space. ...
0
votes
1answer
114 views

On the resemblance of Fubini and integration by parts.

Suppose we have an expression of the follwoing form; $\int_a^b fg $, and want to understand it. By integration by parts we have the following $\int_a^b fg = -\int_a^b Fg^{'} + Fg \mid$. But $\int_a^b ...
6
votes
3answers
109 views

Show that $\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $

I'm trying to show that $$\int_{0}^{+\infty} \frac{\sin(x)}{x(x^2+1)} dx = \frac{\pi}{2}\left(1-\frac{1}{e}\right) $$ using Jordan's lemma and contour integration. MY ATTEMPT: The function in ...
2
votes
3answers
69 views

Why is $\int_{0}^{1}\frac{1}{x} dx = \int_{1}^{+\infty}\frac{1}{x} dx = +\infty$?

I have the theoretical proof but I can't visualise it, doesn't it mean that the area under the function $\frac{1}{x}$ is infinite? How is it possible since it tends towards 0?
1
vote
2answers
99 views

Evaluating $ \int \frac{\sin x+\cos x}{\sin^4x+\cos^4x}\,dx $

$$ \int \frac{\sin x+\cos x}{\sin^4x+\cos^4x}\,dx $$ What I have tried : I tried writing denominator as $ \sin^4x+\cos^4x = 1-2\sin^2x\cos^2x $ and $ 2\sin^2x\cos^2x = \frac{1}{2}\sin^2(2x) $ so ...