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Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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4answers
31 views

$\int_{0}^{2\pi }\cos^{2}(nx)dx$

$$\int_{0}^{2\pi }\cos^{2}(nx)dx$$ Why If I solve the integral by parts I get an answer and if I solve by using formula I get another answer. What's wrong ? By parts: I take $f(x)=\cos^2(nx)$ so $f'...
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0answers
8 views

Good upper bound for $\sum_{n=1}^N a^{-n} n^{-b}$, for $a, b \in (0, 1]$

Let $a, b \in (0, 1]$ and define $S_N:=\sum_{n=1}^N a^{-n} n^{-b}$. Question What is a good upper bound for $S_N$ ? Observations By a simple (and probably careless) application of Cauchy-Schwarz, ...
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0answers
17 views

How to integrate $\int_{-\infty}^{\infty} \frac{e^{i x \beta}}{(x^2+c)^{1/2}} dx $

I try to integrate I tried to integrate $$\int_{-\infty}^{\infty} \frac{e^{i x \beta}} {(x^2+c)^{1/2}}\,\, dx $$ but failed. Does anybody know a way, e.g. by substitution or residue theorem?
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0answers
17 views

Prove integral inequality :

I'm try to prove that : $\int_0^1\int_0^1\int_0^1(\frac{abc}{5}+\exp(\frac{-abc}{4}))dadbdc≤1$ Can use Cauchy Schwartz here or AM-GM Help me or hint me Thanks!
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1answer
39 views

Prove that : $\exists m \in ({\pi/4},{\pi/3})$ such that

I need to prove that $\exists m \in ({\pi/4},{\pi/3})$ such that $$\int_\frac{\pi}{4}^{\pi/3}\frac{1}{x\tan x}dx≤\frac{\pi}{12}\left(\frac{\ln(m)}{m^2}-\frac{\ln(m)}{\sin^2 (m)}+\frac{1}{m\tan m}\...
2
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1answer
18 views

If function has positve integral on domain then there exist interval and positve number such that function is bounded below by that number on Interval

Suppose $\int_a^b f$ exist and positive Prove that there exist interval [c,d] and m>0 such that $f(x)\geq m$. I was thinking to prove by contradiction . Suppose there is no interval with above ...
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0answers
9 views

When Pettis/Dunford integral coincide with Bochner integral?

My question concers Pettis/Dunford and Bochner integral. Assume that $X$ is a reflexive and separable Banach space and $f\colon \Omega\to X$ is weakly measurable and $\int_{\Omega}\|f\| \,d\mu<\...
2
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2answers
25 views

Is $f(x,y)=\frac{x+y}{x^2+2y^2}\sin(e^{xy})$ integrable on set K

Is $f(x,y)=\frac{x+y}{x^2+2y^2}\sin(e^{xy})$ integrable on set $K=\{(x,y): x^2+y^2<=1, x>0, y>0\}$ I am not sure if my argument is viable. Lets take $y=x$ and check if this function is ...
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1answer
24 views

Answer verifcation: $\int\sin(ax)\cos(ax)\,dx$ where “a” is a constant

I was currently doing a trigonometric substitution, and I noticed my answer is not on Wolfram Alpha Answers. This is the integral which I had to solve: $$\int\sin(ax)\cos(ax)\,dx \ \mathbf{\ \ \ \ ...
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1answer
30 views

Understanding the domain of the triple integral for $f(x,y,z)=x+2y+z^2$

So, I am having trouble (again) with the domain for a triple integral of a function, bounded by the paraboloid $2y^2=x$ and the $x+2y+z=4$ and $z=0$ planes I have tried to guess the bounds for x,y ...
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0answers
22 views

Evaluating limit of a function with integral involved.

We need to evaluate the following limit: $$\lim\limits_{x\to 0} \frac{1}{x} \int_0^x \sin^2\left(\frac{1}{y}\right)\,\mathrm dy.$$ I am finding hard to solve this integral as I cannot see a clear ...
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3answers
34 views

Find an example to a function $f$ so $f>0$, $\lim_{x\to\infty} f(x)$ does not exist and $\int\limits_{1}^{\infty} f(x) dx<\infty$

An example I instantly came up with is: If $x\in \mathbb{Z}$ I say that $f(x)=1$ If $x\notin \mathbb{Z}$ I say that $f(x)=\frac{1}{x^2}$ Now, I am pretty sure that this example is correct. However, ...
4
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2answers
114 views

Find : $\int_0^{\pi/4}x\ln(\sin x)\mathrm dx$

I'm try to find this integral $$\int_0^{\pi/4}x\ln(\sin x)\mathrm dx$$ My try use : $\ln(\sin x)=-\ln2-\sum\limits_{n=1}^{\infty}\frac{\cos (2nx)}{n}$ But I don't know how to complete summation ......
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0answers
13 views

Path Integral equals zero on non conservative field

I was doing some excercises and I was asked to compute the line integral along certain path. I used greens formula to calculate the work. When computing the integral I had to divide the domain in two ...
1
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1answer
20 views

Volume of a solid $y=\cos(x)$ and $y=0$ for the interval $0\le x \le \frac{\pi}2$

Volume of a solid $y=\cos(x)$ and $y=0$ for the interval $0\le x \le \frac{\pi}2$. I used method of shells to get: $$2\pi\int_{0}^{\pi/2}x\cos(x)\,dx.$$ And I got: \begin{align} 2\pi\int_{0}^{\...
2
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2answers
33 views

Let $\mu(X)=1$, $0 \leq f \leq k$, and $m=\int_X f d\mu$. Show $\int_X |f-m|^2 d\mu \leq \frac{k^2}{4}$.

Let $\mu(X)=1$ for $\mu$ a positive measure. Let $0 \leq f \leq k$ for some $k\in\mathbb{R}$ and let $m=\int_X f d\mu$. Show $\int_X |f-m|^2 d\mu \leq \frac{k^2}{4}$. My attempt: I tried to expand ...
0
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1answer
37 views

Integrate $\frac{1}{1+n^2x}$

$\int_0^1\frac{1}{1+n^2x}$. Here n is a constant. I know I can do a u-sub and get $\frac{\ln(1+ n^2x)}{n^2}$, but could I also look at it like $\int_0^1\frac{1}{1+(n\sqrt{x})^2 } = \arctan (n\sqrt{...
2
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1answer
40 views

CAS integrals of discontinuous functions

Background This post is motivated by my interest in the performance of symbolic integrators in computer algebra systems (CAS's), such as Mathematica (MMA). I've found that, when an integrand has ...
2
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0answers
12 views

Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions?

Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions? I know that by the definition of an Archmedean sequence of ...
1
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4answers
51 views

Integrating $\int \tan^3(x)$ in two different ways gives two different answers

I was trying to find the antiderivative of a function $$\int \tan^3(x)$$ However, due to substitution differences, my book has a answer of $$\frac12\tan^2(x)+\ln(\cos x)+C$$ while I got an answer $...
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1answer
24 views

Continuity of Lebesgue integral of integrable function

I'm tackling the following question: My approach to (a) was: Let $x \in (0, \infty)$ and consider any sequence $x_j \to x$. We are asked to prove that $v(x_j) \to v(x)$ i.e. that $$\int_\mathbb R \...
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0answers
38 views

Integrating $\left|f(x)\right|$ by pulling out $\mathrm{sgn}(f(x))$ from the integral

I tried doing the following integral: $\int_{0}^{\pi/4}\sqrt{1-\sin2x}\mathrm dx$. Firstly I completed the square by rewriting $1$ as $\sin^2x+\cos^2x$ to get the integral revised to this form: $$I=\...
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votes
0answers
22 views

Inverse Laplace transform of $F(s) = \exp(-a\sqrt{s})/s$, $a > 0$

Show that the inverse Laplace transform of F(s) = $e^{-as^{1/2}}/s$, $a > 0$, is given by $$f(x) = 1 - \frac{1}{\pi}\int^{\infty}_{0} \frac{\sin(a\sqrt{r})}{r}e^{-rx}dr$$ Note that the integral ...
4
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0answers
51 views

Is there a way to preform this integral such that the answer is $e^{-|y|}$?

Consider the function $f(y)=e^{-|y|}e^{y}$ I am trying to integrate this function with respect to another variable (such as $x$) so that the result from the integration is $e^{-|y|}$? The function $...
2
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1answer
95 views

Find $\int_0^{\frac{\pi}{2}} \frac{\sqrt{3-(\cos x)^{2}}}{\sin x + \cos x}dx$

I see this question before in book so i need closed form : $\displaystyle \int_0^{\pi/2}\frac{\sqrt {(3-\cos^2(x))}}{\sin x+\cos x}dx$ I'm trying $t=\tan\frac{x}{2}$ Actually I don't know how I ...
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0answers
41 views

Showing that a certain “norm-like” function fails to satisfy triangle inequality

Let $I$ denote the unit interval. Define $f: I \times I \to \mathbb{R}$ by $$f(x,y) = \begin{cases} 0 & x \in [0, \frac{1}{2}], \ y \in [0, \frac{1}{2}] \\ 0 & x \notin [0, \frac{1}{2}], \ ...
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votes
1answer
27 views

Volume of surface Revolution by area bounded by $y=\sin(x)$ and $y=0$

The revolution across the y-axis, and the bounded area is between $y=\sin(x)$, and $y=0$ for ${0\le x \le \pi}$. I did: $$V_{shells}= 2 \pi \int_\limits{0}^\pi x\sin(x)dx=2\pi^2$$ I am trying to do ...
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1answer
34 views

Let $F(x)= \int^x_0 e^{sint}dt,x \in \mathbb{R}$, then $F'(0)$ equals: [on hold]

$-2$ $-1$ $0$ $1$ $2$ How can I solve this integral?
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0answers
7 views

How to solve $F(t) = a\sin{t} - 2 \int^1_0 F(u) \cos(t - u) du$

How to solve $F(t) = a\sin{t} - 2 \int^1_0 F(u) \cos(t - u) du$ The answer is $F(t) = at e^{-t}$ I am clueless about these types of integrals where we have variables on both sides of equation. ...
0
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2answers
26 views

Integrals of products of sines and cosines with arbitrary periods

I am currently studying the Fourier series, which involves integrals of products of sine and cosine functions. Because sine and cosine are orthogonal, we have been using the following facts to help us ...
1
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0answers
31 views

Understanding the Following Integral Notation

I'm a little confused on the notation my professor used for the following integral. \begin{equation} \int \bar{Y}_{l_f}^{m_f} \left( \dfrac{-Y_1^1 + Y_1^{-1}}{\sqrt{2}}, \dfrac{iY_1^1 + iY_1^{-1}}{\...
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0answers
48 views

How to solve $\int^1_0 \frac{y(u)}{\sqrt t - u} {d}u = \sqrt t $

How to solve $\int^1_0 \frac{y(u)}{\sqrt {t - u}} {d}u = \sqrt t $ Answer is $ y(t) = \frac{1}{2}$ I am clueless about these type of integrals, which have variables on both side of equation, please ...
0
votes
1answer
55 views

Laplace Problem : How to solve this integral?

As far as I know the Laplace transform of $(1/t)$ is infinite. But the problem I have been given is $$\int_{0}^{\infty}\frac{e^{-t} - e^{-4t}}{t}\,dt.$$ How do I solve this problem?
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1answer
23 views

Understanding the domain of the triple integral for $f(x,y,z)=x^2+y^2$

So, I am having trouble with the domain for the triple integral of $f(x,y,z)=x^2+y^2$, bounded by the paraboloid $x^2+y^2=2z$ and the $z=4$ plane I am currently trying to project it on the XY axis, ...
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0answers
22 views

Primitive of a function

I have a function $h_{j}(u_j) = 1-z^2 _j $ with $z_j= \Phi^{-1}(u_j)$, $\Phi^{-1}$ is the standard normal quantile function and $u_j \in (0,1)$ I want to show that $ \int_0 ^{u_j} h_{j} (\lambda) \, ...
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0answers
20 views

Using Archimedes-Riemann Theorem to prove integrability

Here is an excerpt from a book I am reading. There is one portion that I don't understand, which I explain afterwards: For the rectangle $\mathbf{I} = [0, 1] \times [0, 1]$ in the plane $\mathbb{R}^{...
3
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1answer
75 views

Evaluating $\int_{-2}^{-1} \int_{-2}^{-1} \int_{-2}^{-1}\frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz$

I am trying to solve this triple integral problem , but I am having some issues. $$\int_{-2}^{-1} \int_{-2}^{-1} \int_{-2}^{-1}\frac{x^2}{x^2+y^2+z^2} dx dy dz$$ I tried with the 2 different ...
1
vote
1answer
24 views

Evaluate $\iint_{\Sigma} \vec{F}\cdot d\vec{\sigma}$:

$\vec{F}(x,y,z)=(x+1,y-2,z)$ and $\Sigma$ the part of the curved surface of the cylinder $x^2+y^2=2x\,(y\ge 0)$ bounded by the plane $z=0$ and the conical surface $x^2+y^2=z^2\,(z\ge 0)$. The normal ...
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0answers
11 views

Line integral over intersection of a plane and cylinder

Calculate $\int_{\Gamma}\vec{F}\cdot d\vec{s}$, with $\vec{F}(x,y,z)=(x^2,y^3,x^2)$ and Γ the intersection curve of $x^2+y^4=1\,(x\ge0)$ and the plane $x+y+z=1$, oriented from $(0,1,0)$ to $(1,0,0)$. ...
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votes
1answer
20 views

Checking saddle point or not - using rules of 'Fundamental Theorem of Calculus'

True or false For the function, $f(x,y)=\int_{2x}^{-y+2{x^2}}e^{-{t^2}}dt$ $(0,0)$ points is the saddle point. I can do it the long way by solving the integral first but I believe there is a way to ...
2
votes
1answer
46 views

Shells Vs Disks Method: $\ln(x)$

y=$\ln(x)$, and the x-axis, and $x=e$ Rotation across the x-axis: $$V_{shells}=\int_\limits{1}^{e}(\ln(x))^2dx= \pi(e-2)$$ Why is it that when I do the method by shells I have to substract $e-e^x$ $...
0
votes
1answer
41 views

Find the region bounded by $y=x \sin x$, and $y=x$

Find the area bounded by the region $y=x \sin(x)$, and $y=x$, for $0\le x\le \frac{\pi}{2}$. My attempt Area $=\int_\limits{0}^{\frac{\pi}{2}}(x-x\sin(x))dx$ After integrating I got: $$[\frac{x^2}{...
0
votes
5answers
50 views

What is the value of integral $\int \frac{{d}x}{\sqrt {x^2 + 9}}$

What is the value of integral $\int \frac{{d}x}{\sqrt {x^2 + 9}}$ The answer is $\sinh^{-1} (\frac{x}{3})$ I have tried solving it by putting $x = 3\tan \theta$, and got the answer $\ln |sec tan^{-1} ...
0
votes
1answer
43 views

Find the area between $r=a\cos(\theta)$ and $r=a(1+\cos(\theta))$

So, I have to calculate an integral with a domain limited by two functions: $r=a\cos(\theta)$ and $r=a(1+\cos(\theta))$ , where $a>0$ The issue here is that I cannot wrap my head around what the ...
1
vote
1answer
38 views

Poisson equation $-\Delta u = 1$ with Dirichlet condition

Let $R > 0$. Determine the radial solution of the problem \begin{align} - \Delta u(x) & = 1 \text{ if $|x| < R$}\\ u(x) & = 0 \text{ if $|x| = R$} \end{align} We know the fundamental ...
3
votes
1answer
84 views

How can I find the following integral [on hold]

What's the closed form of this integral ?: $\displaystyle\int_{0}^{\pi/2} {\,\sqrt{\, 5 - \cos\left(2x\right)\,}\, \over \sin\left(x\right) + \cos\left(x\right)}\,\mathrm{d}x$ I'm not sur if the ...
1
vote
0answers
37 views

Am I allowed to integrate by parts?

I have the following situation: Consider a function $f\in H^1(\mathbb{R})$, and a uniformly bounded function $S\in\mathcal{C}^\infty(\mathbb{R})$ with well-defined limits at $\pm\infty$ and such that ...
1
vote
2answers
72 views

Show that $\int_0^\sqrt{\pi} e^{-x^2}\sin(x^2)dx>\frac{\sqrt{\pi}}{2}e^{-\pi}$

Please solve it with basic multivariate calculus skills. I attempted multiple times but couldn't prove it without using WolframAlpha. Motivation : note that the RHS bounds from above $$\int_{\sqrt\pi}...
0
votes
2answers
26 views

Finding solution to a non-linear differential equation

I am given the equation below, where b is the average number of births (b=8), and d is the average number of deaths (d=3), and I am given an initial condition: P(0) = 500. Given the above values, I ...
0
votes
2answers
42 views

How to find the integral of $\frac {1}{x^2-x+2}$?

I am learning homogeneous differential equations, and I tried to answer one of the questions, but I am stuck at integrating the equation above. Are there any easier ways than making the denominator: $$...