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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1answer
62 views

how to evaluate $\int_{-\infty}^{\infty} dx\cdot 4x^{2}e^{-2x^2} e^{-a(b-x)^2}$

I have been struggling to evaluate the integral.: $$ \int_{-\infty}^{\infty} dx\cdot 4x^{2}e^{-2x^2} e^{-a(b-x)^2} $$ This is what I did so far. \begin{equation} \begin{split} ...
2
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4answers
39 views

is it true that $\int_0^{\infty} \sin(f(x))dx$ bounded for all continuous $f(x)$(or at least for some of them )?

I'm studying for a test and i see a lot of questions asking to decide if a given integral is bounded or not, many of this integral involve periodic functions like $\sin$ or $\cos$ and since ...
1
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3answers
60 views

Integration for $\int_a^b f(k-x)dx$

Knowing that $\int_a^b f(x) dx=F(b)-F(a)$. What if $\int_a^b f(k-x)dx$ assuming k is a constant which is always greater than x, a, and b?
0
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1answer
17 views

basic knowledge of rate of change

My answer for (a) is that the size of tumor decreases as time goes by. It could possibily because medical treatments are applied. My answer for (b) is the tumor increases at start and decreases ...
1
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3answers
80 views

Is it possible to solve $\int \frac{1}{u^2-u}du$ without hyperbolic tangent?

I'm getting stuck on this integral, and all the tools I see online relate it to hyperbolic tangent. When I try to solve it I break it up using partial fraction decomposition to $\int ...
1
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0answers
24 views

Study the behavior of $\int_1^x \int_1^{t^2}\frac{\sqrt{1+u^4}}{u}\,du\, dt$

Let $$F(x)=\int_1^xf(t)dt$$ $$f(t)=\int_1^{t^2}\dfrac{\sqrt{1+u^4}}{u}du$$ Write expression for $F'(x)$ and $F''(x)$. Determine when $F(x)$ is rising, concave up, has a relative Max or Min. Sketch ...
2
votes
2answers
70 views

Integrating Joint Random Variable Distributions - Defining Integration Intervals

I'm currently studying for a stats quiz using the course text book, and I'm discovering that I'm a little rusty on setting up regions for double integrals. The current question I'm hung up on is as ...
0
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1answer
46 views

Integral of ArcCos with Difficult Argument

I have $$\int_{d-1}^{1}2 x \arccos\left(\frac{x^2+d^2-1}{2 x d}\right)\,\textrm{d} x$$ but can't find the right substitution. I have little experience integrating $\arccos$ with anything but trivial ...
2
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0answers
22 views

Integration of unknown derivative

I am unable to solve this integral, have forgotten basics and so need help. Shall be very thankful If a way out is provided: $\int_0^R \ln[p'(t)]dN(t) - \int_0^R p'(t) dt$ If $p(t)$ was known then I ...
1
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0answers
74 views

Surface Integrals with Cylindrical Coordinates

I'm currently trying to understand cylindrical and spherical coordinates more. I understand them by rote and less by actually understanding what they mean/why we should use them. Right now, I'm ...
4
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2answers
33 views

$f(x) = \int_0^x\frac{1-t^2}{\sqrt{t^4+1}}dt$ find it's derivative and tangent where x = 0

I am given this function: $$f(x) = \int_0^x\frac{1-t^2}{\sqrt{t^4+1}}dt$$ I have to find it's derivative $f'(x)$ and I have to find the equation of it's tangent in the point $x = 0$. I'm a bit ...
0
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0answers
47 views

prove $\int_{o}^{k\alpha}f = 0$

suppose f is an intgrable function on $[0,1] \rightarrow \mathbb{R}$ prove $\int_{o}^{k\alpha} f= 0$. where the integral of $f$ with measure $\alpha$ is zero. $\alpha$ is a number between $0$ and ...
2
votes
1answer
68 views

Convergence test of the following improper integral $\int_0^\infty \frac {e^{-1/x}-1} {\ x^{2/3}}dx$

I've been trying for a couple of hours to prove the convergence of the following integral: $$\int_0^\infty \frac {e^{-1/x}-1} {\ x^{2/3}}dx$$ Eventually I understood from Wolfram-Alpha that the ...
0
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0answers
36 views

$L^2$-function composed with a homeomorphism still $L^2$?

Let $M$ be a compact space equipped with a Borel probability measure $\mu$, let $L^2(M,\mu)$ be the corresponding $L^2$-space, and let $f:M\to M$ be a homeomorphism. Question: If $\varphi\in ...
0
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0answers
43 views

Integral proof $I_n=\int\frac{x^n}{\sqrt{x^2+5}} \, dx$

If $$I_n=\int\frac{x^n}{\sqrt{x^2+a}}\,dx$$ prove that $$I_n=\frac{1}{n}x^{n-1}\sqrt{x^2+a}-\frac{n-1}{n}aI_{n-2}$$ I tried $$I_n=\int\frac{x\times x^{n-1}}{\sqrt{x^2+a}} \,dx =\int(\sqrt{x^2+a})' ...
2
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0answers
89 views

A bit challenging integration. (at least for me its challenging)

Hello everybody I am trying to solve this integral. I show you how far I 've gone. $\displaystyle\int^{\infty}_{-\infty} \frac ...
-1
votes
1answer
35 views

Numerical integration with a step imposed on the antiderivative

Numerical methods, such as Simpson's rule, are well known when you want to integrate with a fixed step $h$ in the independent variable. They have the form of a linear combination of samples of the ...
1
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1answer
37 views

Is it possible to find an explicit form of the solution to $y'=\frac{1-x+y}{x-y}$

We want to solve the differential equation $y'=\frac{1-x+y}{x-y}$, What i did is define $z=x-y$, and then $y'=1-z'$, so overall we have $1-z'=\frac{1-z}{z}$, or in other words $z'=\frac{2z-1}{z}$ ...
3
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1answer
83 views

Help on the following indefinite integral: $\int\big(\sqrt{1-t^2}\big)^{n-1}\mathrm{d}t$

I would like to evaluate the following indefinite integral $$ \int\big(\sqrt{1-t^2}\big)^{n-1}\mathrm{d}t, $$ but -alas- I am not familiarized enough with this kind of integration. I have been ...
0
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1answer
38 views

Proof of expectation of exponential distribution

I need help with understanding the proof of expectation of exponential distribution: I found myself having problems with substituting the limits into $[-xe^{-\lambda x}]$. It probably doesn't make ...
2
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1answer
83 views

Integral of a Dirichlet Series

I'm stuck at a problem of an exercise list... I'd like some help to solve it :) The problem: Suppose that the Dirichlet Series $$A(s)=\lim_{N \to \infty}\sum_{n=1}^Na(n)n^{-s}$$ has abscissa of ...
6
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1answer
148 views

$ \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 $

Evaluate $$I= \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4$$ Answer Options: $1$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{4}$ I need some ...
1
vote
1answer
28 views

Finding measure given constant margins

Suppose $g:[0,1]^2\to R$ and $g$ can have finitely many discontinuities. $F$ is continuous and atomless c.d.f on $[0,1]$ $$\int_{[0,1]} g(x,y)dF(y)=1/2, \forall x$$ $$\int_{[0,1]} g(x,y)dF(x)=1/2, ...
3
votes
2answers
65 views

Help with tricky integral of rational function that deals with positive and negative cases?

Evaluate the integral $$\int \frac{dx}{1+k+(1-k)x^2}$$ for all values of $k \in \mathbb{R}$. Now when $-1 \le k \le 1$ this integral is trivial by using inverse tangent substitution. But this ...
0
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1answer
24 views

Integration without curve formula

My mathematic English is poor, if the question have already been asked with some other therms, I apology I need to process the area of a curve (between two points) which is not drawn from a formula ...
3
votes
2answers
101 views

Why do different substitutions give different results for $\int\cot(x)\csc(x)^2\,dx?$

I am having some issues when trying to integrate this function. First of all I have to decide to make a $U$ substitution either for $\csc(x)$ or for $\cot(x)$, both of them are acceptable ...
1
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2answers
81 views

Cross Product for Biot-Savart Derivation of Current Loop

Biot-Savart's law can be used to determine the magnetic field produced by a figure at a point. Introductory physics texts integrate $dB$ to obtain $B$ where $dB$ = $\frac{I\mu_{0}}{4\pi r^2} * dl ...
4
votes
2answers
82 views

Indefinite integral of $\arctan{\sqrt{1-x^{2}}}$

All is in the title: what is the antiderivative of $x\mapsto \arctan{\sqrt{1-x^{2}}}$ ? I'm supposed to tutor younger students taking an integration class, and this is one of their exercises. I ...
1
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2answers
35 views

Differential calculus: integrate $\frac{1}{x \log^3 (x)}$

I would like a step by step description of how to integrate $$\frac{1}{x \log^3 (x)}$$ I know that the answer is - $\frac{1}{2\log^2(x)}$ and that the integral of $\frac{1}{\log^2(x)}$ is ...
0
votes
1answer
80 views

Problem with Cauchy Principal Value Integral

I'm having problems with this integral $$\int_{a \sqrt \alpha}^\infty \frac{x^2}{(\sqrt{x^2 - a^2 \alpha})(e^x - 1)} \, dx \qquad \text{with } a, \alpha > 0$$ I tried to solve it with Maple, ...
4
votes
2answers
88 views

How to calculate $\int e^{\frac{x^2}{2}} \cdot x^{3} dx$?

$\int e^{\frac{x^2}{2}} \cdot x^{3} dx=?$ I tried to do the substitution $du = x^3dx$, so that $u=\frac{x^4}{4}$. Then, $\int e^{\frac{x^2}{2}} \cdot x^{3} dx= \int e^{\sqrt{u}} du$ However, I would ...
2
votes
1answer
74 views

What are the steps to this functional derivative problem?

I'm trying to derive equations from Matthew Beal's Thesis, Variational Algorithms for Approximate Bayesian Inference pg.55, but I'm stuck on one of the equations (well I'm stuck on a lot of equations ...
0
votes
1answer
26 views

Find p for $\int_{1}^{2}(\frac{x}{(x-1)^p})\,dx$ converges

Find p for $\int_{1}^{2}(\frac{x}{(x-1)^p})\,dx$ converges I tryed this: \begin{align} f(x)& =\int_1^{2} \left(\frac{x}{(x-1)^p}\right)\,\mathrm dx < \lim_{t \to ...
0
votes
1answer
26 views

How to evaluate $\lim_{x\to +\infty} \int_x^{x^3} \frac{dt}{(ln(t))^2}$?

I'm searching $\lim_{x\to +\infty} \int_x^{x^3} \frac{dt}{(ln(t))^2}$, but I'm stuck. I've tried to do a change of variable in order to get $u\to 0$ and then use a Taylor expansion... But nothing ...
1
vote
1answer
78 views

Computing a double integral $\int_{-\infty}^\infty\int_{-\infty}^\infty\frac{f(t)}{1+{(x+g(t))}^2}dt\ dx$

Let $f,g$ be continuous, with $f$ integrable. How can one evaluate $\displaystyle\int_{-\infty}^\infty\int_{-\infty}^\infty\dfrac{f(t)}{1+{(x+g(t))}^2}dt\ dx$ ? Any hint would be welcome. I have ...
0
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1answer
49 views

let $f_n(x) = \frac{x^ n}{n!}$ and $f(x) = \sum f_n(x)$, prove with the Beppo-Levi theorem that:

let $f_n(x) = \frac{x^ n}{n!}$ and $f(x) = \sum f_n(x)$, prove with the Beppo-Levi theorem that: $\int_{a}^{b} f(x) dm(x) = \sum \int_{a}^{b} f_n(x) dm(x)$ if we use Beppo levi, this equation ...
0
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2answers
44 views

Find m for $\int_{0}^{\pi/2}(\frac{1-\cos(x)}{x^m})\,dx$ converges

I need to find all the values of $m$ for which $f(x)$ converges for this function: $$\int_0^{\pi/2}\left(\frac{1-\cos(x)}{x^m}\right)\,dx$$ I tryed this: \begin{align} f(x)& =\int_0^{\pi/2} ...
3
votes
0answers
104 views

Any smart tricks to simplify my nasty integration?

I am trying to solve for the following unpleasant integral $$\int_{\gamma}^{\infty} \bigg[t- \int_{-2}^{2}\frac{ t \ f_X(x)}{1+N \ \big|G(x)\big|^2 \ t^{-3}}\ dx\bigg] \ dt$$ where $N$ is a ...
2
votes
1answer
86 views

How can I solve this integral by hand?

$$\int_0^{85.5}2\pi \cdot 15.537 \arctan\left(\frac x{25}\right)\sqrt{1+ \left(\frac{\frac{15.537}{25}}{\frac{x^2}{625} +1}\right)^2}\,dx$$ I tried using substitution but I couldn't get rid of the ...
3
votes
1answer
72 views

Integration of fraction [closed]

Are there any special cases that make the following true $$\int\frac{f(x)}{g(x)} dx = \frac{\int f(x)\ dx}{\int g(x) \ dx}$$ Thanks
0
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0answers
62 views

How to solve this real-analysis problem?

Since $f(x)$ is bounded we have: $f(x) \le M$ for some $M \in \mathbb{R}$ Also, we have to prove: $$\exists N\in \mathbb{N}\;\;\;\forall n>N\;\;\;:\;\;\; \left| \int_{0}^{1} f(x)\cdot x^n ...
5
votes
2answers
146 views

Definite Integrals problem

The question is to find the value of : $$\frac{\displaystyle29\int_0^1 (1-x^4)^7\,dx}{\displaystyle4\int_0^1 (1-x^4)^6\,dx}$$ without expanding. According to the book, the answer is 7. I tried taking ...
1
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0answers
47 views

Liouville–Hardy theorem: when is $\int f(x) \log(x) dx$ elementary?

I am currently writing a report on Liouville's theorems on integration in finite terms, and I am in the process of proving the Liouville–Hardy theorem. This is what I understand so far. Theorem ...
4
votes
4answers
110 views

Weird integration issue: $\ln(x+1)=\ln(2x+2)$ ?!

Weird integration issue: Using $(\ln[f(x)])'=\frac {f'(x)}{f(x)}$ we get that $\int \frac{2\,dx}{2x+2}=\ln(2x+2)$. Yet, $\int \frac{2\,dx}{2x+2}= \int\frac{dx}{x+1}=\ln(x+1)$ using the same rule as ...
0
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0answers
44 views

The method of undetermined coefficients

What's the proof behind the method of undetermined coefficients that's used in solving second order non-homogeneous differential equation with constant coefficients?
9
votes
1answer
125 views

Theorems or results one can use to prove properties of integrands

Consider a function $f:[a,b]\times \mathbb R \rightarrow \mathbb R$, $(t,x)\mapsto f(t,x)$ for which we do not know much about its regularity in $x$. Define now $$(t,x) \mapsto g(t,x):= \int_a^t ...
1
vote
1answer
83 views

Commutators involving $\Box$ and $\Box^{-1}$

How to determine the followings: $$[\Box,\frac{1}{\Box}]\mathcal{O}=?$$ $$[\nabla,\frac{1}{\nabla}]\mathcal{O}=?$$ $$[\nabla^2,\frac{1}{\nabla^2}]\mathcal{O}=?$$ ...
4
votes
1answer
120 views

Integration over the intersection of the $n$-ball and a hyperplane

Let the $n$-ball of radius $r$, centred at $\mathbf{x}_0$, which will be denoted as the region $$ U = \{\mathbf{x}\in\Bbb{R}^n\colon\|\mathbf{x}-\mathbf{x}_0\|^2\leq r^2\}, $$ and is shown in the ...
0
votes
2answers
30 views

problem with Line integral of vector field

Taking the xyz-coordinate system with $i,j,k$ are the unit vector of each axis, there is a Vector Field $F = {5x+y, 3y-2xz, z} = (5x+y)i + (3y-2xz)j + zk$ I want to find the integral of F on the line ...
0
votes
1answer
213 views

Find the surface area of the portion of the cone $z^2=x^2+y^2$ that is inside the cylinder $z^2=2y$.

(1) I have solved the problem, but I am not sure about the number of octants the surface covers (this affects the final answer value). (2) Also, I have questions regarding the intersection of the ...