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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13 views

Prove that $f(x)$ is differentiable at any point

Given this condition: $x^2sin(x^2) \le x^3f(x)\le sin(x^4)$ at any open interval that goes through $0$ I need to prove that f(x) is differentiable at x=0, but I couldn't come up how, it looks like ...
1
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0answers
22 views

Problem on Integration: $\Bbb R-\Bbb C$ split and pull back of forms

This post is not short. However I'm sure that a guy who good handle these concepts, could read and answer in a five minutes. I only want to write my attempt, in order to understand where I'm wrong. ...
1
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0answers
9 views

Expected value of an expected value of a joint density function

I had a question I was hoping for some help on: Let $Y_1$ and $Y_2$ be continuous random variables with joint density function: $$f(x,y) = \begin{cases} 6(1-y_2) & \text{if $0 <= y_1 <= ...
2
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1answer
45 views

Find $\int_0^1 \frac{dx}{(1+x^n)^2\sqrt[n]{1+x^n}}$

Find $$\int_0^1 \frac{dx}{(1+x^n)^2\sqrt[n]{1+x^n}}$$ with $n \in \mathbb{N}$. My tried: I think that, needing to find the value of $$I_1=\int_{0}^1 \frac{dx}{(1+x^n)\sqrt[n]{1+x^n}}$$ because: ...
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3answers
54 views

A level Integration

Using the substitution $x=\cosh (t)$ or otherwise, find $$\int\frac{x^3}{\sqrt{x^2-1}}dx$$ The correct answer is apparently $$\frac{1}{3}\sqrt{x^2-1}(x^2+2)$$ I seem to have gone very wrong ...
3
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2answers
48 views

Convergence testing of the improper integral $\int_{0}^{\infty}\frac{\ln x}{\sqrt{x}(x^2-1)}\ dx$

I've tried to test this integral for convergence for a couple of hours, actually I know that $$\int_{2}^{\infty}\frac{\ln x}{\sqrt{x}(x^2-1)}\ dx$$ converges with no problem with the help of Dirichlet ...
1
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1answer
28 views

Surface/Path Integral Approach - Brain Fart?

Many times I have dealt with path and surface integrals of the following form $$\int_C \mathbf{F}\cdot d\mathbf{r} \,\,\,\,\,\textrm{(path integral)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_S ...
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1answer
23 views

How to compute $\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}\:\text{ for }\: a>0, b\in \mathbb{C}$?

As the title says I am trying to compute the integral $I=\displaystyle\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}$ where $a>0$ and $b$ is a complex number. For the special case of $b=-2a$, we ...
1
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2answers
35 views

Surface Area by Integration

$$2\pi\int_{3}^6\left(\frac{1}{3}x^\frac{3}{2}-x^\frac{1}{2}\right)\left(1+\left(\frac{1}{2}x^\frac{1}{2}-\frac{1}{2}x^\frac{-1}{2}\right)^2\right)^\frac{1}{2}dx$$ I've managed to simplify this down ...
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2answers
103 views

Integration help - question: $e^{-\sin(x)}$

I would really like some help with the integration of $e^{-\sin(x)}$. Thanks to anyone who will help :) Given that $\sin(x) > \frac{2x}{\pi}$ for $0 < x < \frac{\pi}{2}$, where ...
1
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1answer
81 views

Volume of figure between $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ if $z\geq 0$

I have a problem where I have to find volume of figure formed, when $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ intersects if $z\geq 0$. Here is a graphic for clarity: So far I have transformed the problem to ...
3
votes
2answers
50 views

Find $\lim \limits_{x \to \pi}\frac{\int_0^x\cos^2(t)dt}{x-\pi}\;$

$$\lim \limits_{x \to \pi}\frac{\int_0^x\cos^2(t)\,dt}{x-\pi}$$ I don't understand why the limit is not $\infty$ How is the limit: $1$?
37
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9answers
2k views

Closed form for $\int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$

I've been looking at $$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$ It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example: ...
1
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1answer
41 views

How to show $\int_0^{\infty} \frac{1}{\sqrt{x}}\sin({\frac{1}{x}})dx$ converges

How to show $\int_0^{\infty} \frac{1}{\sqrt{x}}\sin({\frac{1}{x}})dx$ converges? I have that $$\frac{-1}{\sqrt{x}}\le \frac{\sin({\frac{1}{x}})}{\sqrt{x}} \le \frac{1}{\sqrt{x}}$$ but when you ...
8
votes
1answer
143 views

A fractional part integral giving $\frac{F_{n-1}}{F_n}-\frac{(-1)^n}{F_n^2}\ln\left(\!\frac{F_{n+2}-F_n\gamma}{F_{n+1}-F_n\gamma}\right)$

I've been asked to elaborate on the following evaluation $$ \begin{align}\\ \displaystyle {\large\int_0^{1}} \!\cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots + \cfrac 1 { 1 + \psi ...
0
votes
1answer
21 views

General question about simplification

After done with integration I got the final answer as: $\ln(a+4) + \ln(a-4) + C$ I can rewrite it as: $\ln((a+4)(a-3)) + C$ But in book it is written as: $\ln((a+4)(a-3)+C)$ Is it correct? and ...
3
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1answer
27 views

Convergence of series of integrals

Let $\phi \in C^\infty(\mathbb R)$ be a function such that $\phi(x), \phi'(x) \to 0$ as $x \to \infty$. I want to show that $$\lim_{n \to \infty} \int_\mathbb R \cos(nx) \phi(x) \ dx = 0$$ Doing it ...
6
votes
3answers
175 views

A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1(x)_n dx$

I would like to obtain a numerical evaluation of the series $$S=\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x(x+1)\cdots(x+n-1)\: dx$$ to five significant digits. I've used ...
0
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0answers
19 views

Solving System of 2 simple odes

I am just trying to solve two simple odes using Runge-Kutta method: \begin{equation} \frac{dx}{dt} = v \end{equation} \begin{equation} m .\frac{dv}{dt}= f_{1}(x)+f_{2}(x,v) \end{equation} ...
1
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1answer
46 views

Find $\lim \limits_{x \to 0} \frac{\int_0^x \frac{t\,dt}{\cos t}}{\sin^2(x)}\,$

$$\lim_{x \to 0} \dfrac{\int_0^x \frac{t\,dt}{\cos t}}{\sin^2(x)}$$ what does it mean when the limit of $x$ is $0$ in the integral? How do I calculate this limit?
3
votes
1answer
41 views

Need help with continuing an idea concerning showing that $4\sum\limits_{n \ge 1} a_n^2 \ge \sum\limits_{n \ge 1} \frac1{n^2}(a_1+…+a_n)^2 $

I recently encountered the following problem: If $\sum a_n^2 $ converges and $\alpha_n= \frac{a_1+...+a_n}{n}$ then show that: $$4\sum_{n \ge 1} a_n^2 \ge \sum_{n \ge 1} \alpha_n^2$$ I had an ...
1
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0answers
22 views

Proving that $\int_{\mathbb{R}} f \ d\mu = \frac{1}{N}\sum_{i=1}^N f(\lambda_i)$

I want to know if my proof is correct and if there is some easier way to prove this (you don't need to read all my proof, I'm accepting as answers another proofs, not just corrections of mine). ...
1
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2answers
37 views

Finding the value of $3(\alpha-\beta)^2$ if $\int_0^2 f(x)dx=f(\alpha) +f(\beta)$ for all $f$

Let $f$ be a polynomial of degree $n$ at most $3$ such that there exists some $\alpha,\beta$ satisfying $\int_0^2 f(x)dx=f(\alpha) +f(\beta)$ for all such $f$. Find the value of $3(\alpha-\beta)^2$ ...
0
votes
0answers
51 views

Integration over a variable

Can someone explain to me the step by step of this integration? $$∫_0^r(a-\frac{r}{b})dv$$ Where $v$ is the volume of a cylinder $ \pi r^2h$ The answer is $$ \frac{a}{2}-\frac{r}{3b}$$ But it's ...
3
votes
2answers
48 views

Evaluating the Definite Integral $\int_0^{\pi}\cos^{2n} \theta d\theta$

$$\int_0^{\pi}\cos^{2n} \theta d\theta$$ $$u=\cos \theta \implies du= -\sin \theta d\theta \implies d\theta= -\frac{du}{1-u^2} $$ $$\int_{-1}^1 \frac{u^n}{1-u^2} du=\int_{-1}^1 ...
1
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2answers
29 views

Find $\int (e^{2x}+e^{3x})^\frac{1}{2}dx$

$$\int (e^{2x}+e^{3x})^\frac{1}{2}dx$$ I'm not sure what substitution I'm supposed to make here. Can someone help?
1
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3answers
55 views

$\int^\infty_0 \frac{\cos(x)}{\sqrt{x}}\,dx$ Evaluate using Fresnel Integrals

$\int^\infty_0 \frac{\cos(x)}{\sqrt{x}}\,dx$ Evaluate using Fresnel Integrals (For reference the $\cos$ Fresnel integral is $\int^\infty_0 \cos(x^2)\, dx = \frac{\sqrt{2 \pi}}{4}$) I've tried ...
1
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1answer
20 views

Are these two elliptic integral evaluations identical?

I'm reading a paper on the Schwarz D minimal surface, and I'm wondering whether the authors have made a mistake. They evaluate the integral $$ \int_0^z \frac{2t\;\mathrm{d}t}{\sqrt{t^8-14t^2+1}}, $$ ...
1
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1answer
30 views

How to find $\int_0^{1/4}\frac{1}{x\sqrt{1-4x}}\ln\left({\frac{1+\sqrt{1-4x}}{2\sqrt{1-4x}}}\right)dx$

Let $H_n$ be the harmonic series. I want to find the value of $A=\displaystyle\sum_{n=0}^\infty \binom{2n}{n}\left(\frac{1}{4}\right)^n\frac{H_n}{n} $. From this paper : ...
0
votes
1answer
747 views

proof of the second generalized mean value theorem for integrals

Let $f,g,g´$ be continous on $[a,b]$ and $g$ monotone on $[a,b]$; then there exist $c\in (a,b)$ so that $$\int_{a}^{b}f(x)g(x)dx=g(a)\int_{a}^{c}f(x)dx+g(b)\int_{c}^{b}f(x)dx$$ Ineed to apply the ...
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1answer
104 views
+50

analytic solution of a definite integral

Integrate the following $$\int_0^\infty \alpha\,\beta\, c\, k\, x^s\, x^{c-1} (1+x^c)^{k-1} \left[(1+x^c)^k-1\right]^{-\beta-1} \left[1+\gamma ...
1
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1answer
57 views

Solving integral $\int \arcsin x \cos x dx$ [on hold]

Can anyone give me a hint how to solve $\int \arcsin(x)\cos(x)dx $ ?
0
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1answer
27 views

What does it mean $\int_1^\infty\frac{F(y)}{y^2}\mathrm dy$?

Which type of functions will satisfy this? $$F: [1,\infty) \to [0,\infty)$$ $$\int_1^\infty \frac{F(y)}{y^2} dy \leq 1$$
8
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3answers
654 views

Compute $\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space$

How would you approach $$\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space?$$ The way I see here involves Dirichlet kernel. I wonder what else can we do, maybe some easy/elementary approaching ...
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1answer
15 views

How to plot function of three or more variable?

How to plot function of two or more variable ? Also,why do we require perpendicular axis for the function to be examined ?
2
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2answers
83 views

Solving an integral (with substitution?)

For a physical problem I have to solve $\sqrt{\frac{m}{2E}}\int_0^{2\pi /a}\frac{1}{(1-\frac{U}{E} \tan^2(ax))^{1/2}}dx $ I already tried substituting $1-\frac{U}{E}\tan^2(ax)$ and ...
2
votes
1answer
268 views

Function zero almost everywhere if $\int fg=0$

Assume $f$ is an integrable function on $\mathbb{R^n}$. Assume for every bounded continuous function g on $\mathbb{R^n}$, $\int_\mathbb{R^n}fg=0$. Prove $f$ must equal $0$ almost everywhere. I am ...
3
votes
2answers
49 views

How can I show that $f$ must be zero if $\int fg$ is always zero?

Let $f(x)$ be continuous on $[a,b]$ and suppose $\int_a^b f(x)g(x)dx = 0$ for every continuous function $g$ on $[a,b]$. Prove that $f(x)=0$ on $[a,b]$. I understand that $f(x)$ must be zero ...
2
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2answers
56 views

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$??

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$? According to Serge Lang, the integral on the left is the error term for Stirling's ...
3
votes
1answer
42 views

How to use U substitution for the integral $\int\frac{8}{49+x^2}\,dx$?

So, the following is my problem. $$\int\frac{8}{49+x^2}\,dx$$ I understand this. I should first take out the constant which is 8 so it'll be $$8\int\frac{1}{49+x^2}\,dx$$ Then I should factor out the ...
13
votes
4answers
298 views

How to prove $\int_0^1\frac{x^3\arctan x}{(3-x^2)^2}\frac{\mathrm dx}{\sqrt{1-x^2}}=\frac{\pi\sqrt{2}}{192}\left(18-\pi-6\sqrt{3}\,\right)$?

How to prove the following result? $$\int_0^1\frac{x^3\arctan x}{(3-x^2)^2}\frac{\mathrm dx}{\sqrt{1-x^2}}=\frac{\pi\sqrt{2}}{192}\left(18-\pi-6\sqrt{3}\,\right)$$ For my part no idea?
0
votes
1answer
27 views

Integration Convergence/Divergence Questtion

$$ \int\limits_0^{\pi} \frac{ dt}{\sqrt{t} + \sin t }$$ How can one tell if this integral converges or diverges? Integral of $1/(\sqrt{t}+\sin(t))$ from $0$ to $\pi$. I can't even find the ...
0
votes
2answers
291 views

Joint distribution of U = X + Y and V = X - Y

I have two independent continuous random variables, X and Y, which are uniformly distributed over the interval [0,1]. From this I have two further random variables, U and V, which are defined as U = X ...
0
votes
1answer
34 views

Prove that $\int_0^{\pi} \sin^nx\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cos(n+2)xdx=0$

Prove that $$\int_0^{\pi} \sin^nx\cdot\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cdot\cos(n+2)xdx=0$$ with $n \in \mathbb{N}$ I think it's true, but I can't prove.
1
vote
1answer
43 views

Evaluate an integral involving tangent and secant: $\int \tan^2x\sec^2x\,dx$

Evaluate $\displaystyle \int \tan^2x\sec^2x\,dx$ I tried several methods: First method was I changed $\tan^2x = \sec^2x-1$, and then substitute $\sec x$ to $t$, but it doesn't work. Second ...
2
votes
1answer
18 views

Integration by parts $\int(x+y)e^{-x}dx$

What I'm trying to solve: $\int(x+y)e^{-x}dx$ Here's my professor's approach: $$u = x, du = e^{-x}$$ $$du = dx, dv = -e^{-x}$$ By doing parts: $(-xe^{-x}) - \int(-e^{-x})dx - ye^{-x} = (-xe^{-x}) ...
0
votes
2answers
20 views

Integration inequality question help: Sketch the curve y=1/u for u > 0…

Sketch the curve $y=\frac{1}{u}$ for $u > 0$. From the diagram, show that $\int\limits_1^{\sqrt{x}}\frac{du}{u}< \sqrt x-1$, for x > 1. Use this result to show that $0 < \ln(x) < ...
4
votes
3answers
86 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
3
votes
1answer
48 views

Why does the integral equal $1$?

Let $a\in\mathbb{R}-\mathbb{Z}$. Why is the following equality true? $$1 = \frac{1}{2\pi} \int_0^{2\pi} \left| e^{-i(\pi-x)a} \right|^2 dx$$ More precisely, why is the integrand equals $1$?
3
votes
1answer
28 views

Why does $\int_b^{b+\Delta b}f(x)\;dx=f(b)\Delta b+\mathcal{O}(\Delta b^2)$

On this page it is shown that: $$\frac{\partial}{\partial b}\left(\int_a^bf(x)\;dx\right)=\lim_{\Delta b\rightarrow 0}\frac{1}{\Delta b}\int_b^{b+\Delta b}f(x)\;dx=\lim_{\Delta b\rightarrow ...