Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
70,131
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Integral inequality on Bourgain Spaces
I'm studying the section about Bourgain spaces of Terence Tao's book "Nonlinear Dispersive Equations: Local and Global analysis".
I'm trying to understar the proof of Lemma 2.11 and I'm ...
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2
votes
1
answer
147
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How do we show $\int_{[0,\:1)^2}\exp\left(-\frac{\left\|x-\frac{x_i+x_j}2\right\|^2}{\sigma^2}\right)\:{\rm d}x=\pi\sigma^2$ here?
Let $D=[0,\sqrt N)^d$ for some $N\in\mathbb N$, $x_i,x_j\in D$ and $\sigma>0$. In equation $(30)$ of this paper I've read that, if $d=2$, then \begin{equation}\begin{split}\int_D\exp\left(-\frac{\...
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Contour Integral of $\sin(ax)/(b^2+x^2)$
I am trying to solve this integration.
$$∫\frac{\sin(ax)}{b^2+x^2}\ dx$$
The limits of integration between $0$ to $c$ where $a, b,$ and $c > 0$ ... any idea.
I tried the different similar methods ...
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6
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3
answers
105
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Evaluating $\int \sqrt{\frac{x^2+1}{x^2(1-x^2)}}dx$
Today I came across the following integral
$$\int \sqrt{\frac{x^2+1}{x^2(1-x^2)}}dx$$
Here's my work:
$$\begin{align}\int \sqrt{\frac{x^2+1}{x^2(1-x^2)}}dx& = \int\frac{1}{x} \sqrt{\frac{1+x^2}{1-...
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0
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Integral of vector function && inequality
In this inequality:
$-2x^{T}(t)PA_d\int_{-d}^{0}Ax(t+s)ds \\
\leq d\beta x^{T}PA_dA_d^{T}Px(t)+\frac{1}{\beta}\int_{-d}^{0}x^{T}(t+s)A^{T}Ax(t+s)ds $
I know it comes from the well-known inequality ...
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1
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42
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Is the antiderivative of a Riemann integrable function continuous? [duplicate]
I’m trying to figure out a proof that if $g(x)$ is the antiderivative of $f(x)$ where $f$ is Riemann integrable then $g$ is continuous.
In other words: if $ \int_{0}^{x}f \space dt = g(x) $ and $f$ is ...
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1
vote
1
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Proof of this "fact" on the wikipedia for error function?
Given a random variable $X ~ Norm[μ,σ]$ (a normal distribution with
mean $\mu$ and standard deviation $σ$ and a constant $L < μ$ :
$$ \begin{align} \Pr[X\leq L] &= \frac12 +
\frac12\...
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3
votes
0
answers
17
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Understanding an the last step in proving the Poincaré inequality for smooth functions on $\mathbb{R}^n$.
I'm looking at the following: Let $u:X \to \mathbb{R}$ be an integrable and smooth function on a subspace $X \subseteq \mathbb{R}^n$ equipped with the Lebesgue measure $\lambda_n$, and let $B \...
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0
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0
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38
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Can I simplify this simple equality? [closed]
I am trying to understand a proof and do not understand how they simplified the following equation from :
$\int_{x}^{}x*(something)dx = \int_{z}^{}z*(somethingelse)dz$
to :
$\int_{x}^{}(something)dx = ...
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0
votes
2
answers
45
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How to find the area of a strip from a cone [closed]
Let's say we have a cone with radius height $h$ and radius $r$. Therefore the lateral side, let's call it $\ell$, would be $\sqrt{h^2 + r^2}$. If I want to find the area of the strip/region from $(\...
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0
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Calculus operators in laplace space?
I was playing with laplace transforms and found something curious.
Suppose we have an expression $4x+3$ that we want to take the derivative of or the integral of. If we instead took its laplace ...
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0
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27
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Sufficient conditions for $\mathscr{L}^p$ convergence? [duplicate]
Let $p \in [1,\infty)$, $f,f_1,f_2,...\in \mathscr{L}^1(X,\mathscr{A},\mu, \mathbb{R})$ are all nonnegative functions, such that ${f_n}\rightarrow f$ a.e. It is known that this is not sufficient to ...
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1
answer
107
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What contour to use for Cauchy residue theorem on integral of a rational function with multiple poles?
What contour should I choose to apply Cauchy residue theorem to calculate the integral $$\int_0^\infty \frac{x^{k-1}}{(x+a)(x+b)^{k+1}}dx,$$ where $k$ is a positive integer and $a,b>0$? I tried ...
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63
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How can we sum $\sum_{n=1}^\infty \frac{1}{n^n}$ exactly? [closed]
How can we sum $\sum_{n=1}^\infty \frac{1}{n^n}$ exactly? For that matter what about the related integral $\int_1^{\infty}{dx \over x^x}$?
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1
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1
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If $f$ is holomorphic on the closed unit disc, prove $\int_Cf(z)\log(z)dz=2\pi i\int_0^1f(x)dx$ where $C$ is the unit circle.
If $f$ is holomorphic on the closed unit disc, prove $\int_Cf(z)\log(z)dz=2\pi i\int_0^1f(x)dx$ where $C$ is the unit circle.
Hint is to use integration by part but I can't find a good reference for ...
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2
answers
50
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I need help with integrating an expression with multiple $x$ and $y$ terms [closed]
The equation is $x^2+y^2 = 3\sqrt{2} x - 5\sqrt{2} y +2xy$. I need to try and find the area under the curve and above the x axis between $x = 0$ and $x = 3\sqrt{2}$. I've heard that implicit ...
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1
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52
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Can you explain how to solve this probability with poisson distribution? [closed]
A random variable X is said to have a Poison distribution if given k ∈ ℕ,
px (k) = (e^(-m) * m^k)/k! where m is the expected value of X.! Suppose the amount of students who register for a certain ...
0
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2
answers
47
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Defining formulas for first-order linear differential equations.
When defining the formulas for the first-order linear differentiable functions we are necessitated to define a equation that satisfies $u'(x)$ = $u(x)p(x)$ so then the product rule can be applied. And ...
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3
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48
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Is this weird function with argument in the integrand continuous? (Fundamental Theorem of Calculus)
Let $f:I \rightarrow \mathbb{R}$ with $I$ interval and $f \in C^\infty(I)$. If $t_0 \in I$, we know by the FTC that $F:I \rightarrow \mathbb{R}$ given by $F(x)=\int_{t_0}^x f(t)dt$ is continuous. But ...
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2
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65
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Solve an integral by bringing quadratic trinomial to canonical form
The problem is the following:
$$\int{\sqrt{4x^2+x}dx}$$
Now once gotten to canonical form of a quadratic trinomial, $ax^2+bx+c=a(x-(\frac{-b}{2x}))^2-\frac{b^2-4ac}{4a}$, the intregral looks like this:...
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0
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15
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visualization of triple integrals with iterated bounds in different coordinate systems
Lately i've been trying rather hard to write a code in cpp or python that not only calculates the result of a given triple integral, but also depicts the shape of the volume defined by the integral. ...
2
votes
2
answers
167
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Find $\int\frac{1}{2\sin (x)+3\cos (x)+1}$ $\Tiny{dx}$
Question
Evaluate the following integral: $\int\frac{1}{2\sin (x)+3\cos (x)+1} \small{dx}$
Now, I've tried a couple of different substitutions and integrating partially but unfortunately, to no ...
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0
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27
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Integration by Parts Table Technique With Exponential and Polynomial Higher Order
I come to you with a rather simple question in need of a reference or two from more knowledgable sources. Here was the simple integration by parts problem:
However, a rather peculiar table (book-...
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1
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40
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Need help solving a wind power problem involving wind speed, energy production, and optimization
I am working on a wind power problem and need some help. Here is the problem:
During a windy day, the wind speed at a wind turbine can be described by the model:
$$
v(x)= 11 \sin(0.11x - 0.89) + 28, \...
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2
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148
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Help needed to calculate the derivative of $\int_0^t \cos(x^2) d x$
I'm currently working on a calculus problem and I could use some help. The question is to find the value of $F^{\prime}(\sqrt{\pi})$ if $F(t)=\int_0^t \cos \left(x^2\right) d x$.
I'm not sure how to ...
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63
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What techniques can I use to solve $ \int$ $ \small{\dfrac{2}{t^2(a-b)+2ct+(a+b)} \space dt}$?
I am trying to calculate the following integral:
$$\int \dfrac{2}{t^2(a-b)+2ct+(a+b)}dt$$
I've been going through an old university textbook (at the age of $44$) just to see if I fully understand ...
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1
answer
112
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Integrating $y' = \sin^2 y$
I'm trying to solve: $$y' = \sin^2 y$$
It is a separable variable differential equation, so I arrive to
$$\int\frac{dy}{\sin^2 y}=\int dx$$
I use the identity $$\sin^2 y=\frac{1-\cos(2y)}{1}$$ and ...
0
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0
answers
47
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Difference between $\int$ and $\int_{-\infty}^{\infty}$ [duplicate]
I was browsing questions and I found this one which uses a definite integral from negative infinity to positive infinity. I then wondered if there was a difference between that and the indefinite ...
0
votes
1
answer
23
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Rephrasing Double Integral Equation
Suppose we have a equation $$\iint_{-\infty }^{\infty} xy \frac{1}{2\pi \sigma^2}\exp\left(-\frac{(x-\mu_x)^2+(y-\mu_y)^2}{2\sigma^2}\right) dx dy$$
What property of the integral has been used to ...
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0
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23
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A indentity apply mean value theorem to gradient difference [closed]
I met the following identity when reading a paper:
\begin{equation}
\int_{0}^1\nabla f(x+s\delta)-\nabla f(x){\rm d}s=\left\langle\int_0^1(1-s)[\nabla^2f(x+s\delta)-\nabla^2f(x)]{\rm d}s, \delta\right\...
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1
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77
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Find the complex Fourier transform of $f(x) = \dfrac{1}{1 + x^2 + x^4}$.
How do I find the complex Fourier transform of $f(x) = \dfrac{1}{1 + x^2 + x^4}$?
I know that the complex Fourier transform is $\hat{f}(k) = \displaystyle \int_{-\infty}^{\infty} f(x)e^{-ikx}\ dx$.
...
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1
answer
56
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Show the existence of a limit of an integral
I was presented the following problem at Calculus class:
Let $f: (0, \infty) \to \mathbb{R}$. We know that $f$ is Riemann integrable in every compact interval of $(0, \infty)$, and that the limit $\...
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Integration signum of a continuous function
Let $f:[a,b]\longrightarrow \mathbb{R}$ be continuous, and denote by $\mathrm{sgm}(x)$ the signum of $x$. Define $g(x):=\mathrm{sgm}(f(x))$ for all $x\in [a,b]$. We assume we don't know the expression ...
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1
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Double Integral Split Rule
Suppose we have a function:
$\iint_{-\infty }^{\infty} (ax+by)f_{xy}(x,y) dx dy$
I would like to know what property of the integral rule has been used to rephrase the above equation into like below:
$...
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0
answers
44
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How to find the surface area of the spout of a teapot?
I am trying to find the surface area of a teapot. I found the surface area of the body of the teapot by taking a picture of it, graphing the picture and then calculating the surface area of revolution....
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0
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75
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Find the volume below $z = \sqrt{1 - r^{2}}$ and above the top half of the cone $z = r$
I have made an attempt at this question:
Find the volume below $z = \sqrt{1 - r^{2}}$ and above the top half of the cone $z = r$
I solved for $r$ first, then I calculated:
\begin{equation}
\int_{0}^{...
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0
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An integral problem that has been bothering me for a year
This is the triple integral, the lambda here is going to 0+, the integration result is as follows, I hope someone can help me to answer, thank you very much，enter image description here
enter image ...
2
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1
answer
71
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How does log come into play?
$$\begin{align*}I & = 2\int_{0}^{\frac{1}{\sqrt{2}}} \dfrac{\sin^{-1} x}{x}\ dx - \int_{0}^{1} \dfrac{\tan^{-1} x}{x} \ dx \\ & = 2\int_{0}^{\frac{\pi}{4}} \dfrac{\theta \cos \theta}{\sin \...
1
vote
2
answers
96
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Definite integral over an infinite product
Evaluate the following integral $$\int_0^\infty\frac{x+1}{x+2}\cdot\frac{x+3}{x+4}\cdot\frac{x+5}{x+6}\cdots dx$$
When I saw this, I was pretty sure that the infinite tern must telescope or it must ...
1
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0
answers
21
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Centre of Mass in x,y plane
Given an area $R$ in x,y plane, and the density is $\rho(x,y)$ at $(x,y)$, then we can use double integrals to calculate the centre of mass coordinates. Now my question is, intuitively, given any non-...
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Can you solve this hard integral question? Any ideas? [closed]
For each $n\in\mathbb N$ calculate $\int x^n e^{-x^2} dx$. The answer may be expressed in terms of the function $\operatorname{erf}(x)$ which satisfies $\operatorname{erf}'(x)=\frac2{\sqrt\pi}e^{-x^2}....
0
votes
2
answers
46
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Why don't the bounds in this definite integral change?
The question
This is probably a very basic question but I'm having a brain lapse and don't know why they didn't change the definite integral bounds from ($0 \rightarrow4$) to ($4 \rightarrow20$). I ...
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votes
1
answer
36
views
Limits of Riemann sums left and mid endpoint rule.
In my calculus class we have begun talking about integrals. In particular we have begun talking about Reimann sums and how through the limit of a Reimann sum we can integral. But so far all our ...
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3
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4
answers
161
views
How do I solve $ \int_{0}^{1}\frac{x^2}{\sqrt{3+x^2}}dx$?
I had to solve this integral:
$$
\int_{0}^{1}\frac{x^2}{\sqrt{3+x^2}}\mathrm{d}x
$$
by the substitution $x=\sqrt{3}t$ the indefinite integral can be written as:
$$
\int \frac{x^2}{\sqrt{3+x^2}}\;\...
- 541
1
vote
0
answers
20
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Second Order Homogenous ODE solve with Real Analysis Integration topics [duplicate]
Question:
Suppose that $u\in C([a,b])$ is twice continuously differentiable, $V\in C([a,b])$, $V(x)\geq0$ for all $x \in [a,b]$ and
$$
-u''(x) + V(x)u(x)=0, \;\; x\in [a,b],
$$
$$
u(a)=u(b)=0
$$
Prove ...
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2
votes
2
answers
52
views
If $f$ is $L^1$ and continuous, then does $\int |f(a-x)-f(a)|dx<\infty$ hold?
Let $f:\mathbb R\to \mathbb C$ be a $L^1$ and continuous function, and $a\in\mathbb R.$
Then, does $$\int_{-\infty}^\infty |f(a-x)-f(a)|dx<\infty$$ hold ?
Now, I have $\int_{-\infty}^\infty |f(x)|...
- 2,649
4
votes
0
answers
59
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Stokes Theorem on manifolds with dense corners
I am currently working on a project, where I would (ideally) like to apply Stokes theorem on a Manifold with corners. I have found various sources, which justify this application. Except one thing:
In ...
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2
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0
answers
109
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Evaluate the following trigonometric integral with exponential function
Find the value of $$\int_0^\pi\mathrm{e}^{\mathrm{e}^{\cos\left(x\right)}}\cos\left(\sin\left(x\right)\right)\cos\left(\mathrm{e}^x\sin\left(\sin\left(x\right)\right)\right)dx$$
How to solve this ...
2
votes
0
answers
38
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Integration of Cahn-Hilliard-Oono equation
I am currently reading a paper on the Cahn-Hilliard-Oono equation with Neumann boundary conditions:
\begin{align}
\frac{\partial u}{\partial t} + \epsilon u &+ \Delta^2 u - \Delta f(u) = 0, \quad \...
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0
answers
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Is there a way to show multtiple integrals without using muliple integral symbols or the I symbol? Also can we say the -1 derivative is the integral?
How do I show multiple integrals (20th integral)?
Can I use the -1 derivative to show an integral?
Can I use the -1 integral ( if its symbol exists) to show a derivative?
