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.

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Finding a primitive of a function on some closed interval $I$.

Find a primitive of $f(x):=x$. First, we need to decide the interval on which we find a primitive of $f$. I decide to find a primitive of $f$ on $I=[-1, 1]$. I will find a primitive of $f$ by ...
佐武五郎's user avatar
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1 answer
9 views

Equality of probability measures from given integral

Let $(X,\mathcal{A})$ be a measurable space. Consider $\Bbb{R}$ equipped with its Borel sigma-algebra, and let $\mu,\nu$ be probability measures on $X\times\Bbb{R}$, with the product sigma-algebra. ...
geodude's user avatar
  • 8,065
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1 answer
29 views

Integral over the Riemann sphere

I was reading this wikipedia article and stuck to show that, $$\frac{i}{\pi}\int \frac{dz\wedge d\bar z}{(1+|z|^2)^2}=2$$ after switching to polar coordinates. By Stokes' theorem, an exact form would ...
falamiw's user avatar
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-1 votes
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Show that if $f$ is convex then $g(x)=\dfrac{1}{2h}\int_{x-h}^{x+h}f(t)dt$ is convex.

Let $f: \mathbb{R} \rightarrow \mathbb{R} $ a convex and continuous fonction. Let $g(x)= \dfrac{1}{2h}\displaystyle\int_{x-h}^{x+h}f(t)dt$. Show that $g$ is convex. I showed that $g'(x) = \dfrac{f(x+h)...
Lasky's user avatar
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1 answer
16 views

Expected value of the random variable: $\int (\mu + \sigma Z)\phi(x)dZ$

There is a random variable $X$ and its standard normal viariable $Z = \dfrac{X-\mu}{\sigma}$. I'm looking for an expected value of the random walk defined by $V = \mu + \sigma Z$. For $X \in [0, \...
bag_dush's user avatar
  • 105
2 votes
0 answers
50 views

A common technique in number theory to evaluate integration

Sorry this question is a bit weird ... I saw a technique in some number theory book (that I don't remember the name - I checked Titchmarsh but I couldn't find) that in order to evaluate some integral (...
Ali's user avatar
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48 views

How to compute $\displaystyle\int_{i\infty+1}^1\frac{e^{i2(n+1)x}}{\sin(x)}\, \mathrm dx$

I need to solve $$\int_{i\infty+1}^1\frac{e^{i2(n+1)x}}{\sin(x)}\, \mathrm dx$$ which appears in the closed form of $$\sum_{k=0}^n\frac{e^{i2k}}{2k+1}.$$ By the looks of the integrand, Euler's ...
Joan S. Guillamet F.'s user avatar
1 vote
1 answer
41 views

Integrate a sum of trig function under absolute value

Let $n \in \mathbb{N}$, I'm trying to compute en explicit formula for the following integral: $$ \operatorname{I}\left(n\right) = \int_{\left[0,2\pi\right]^{n}} \left\vert\cos\left(x_1\right) + \cdots ...
MathRevenge's user avatar
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Volume of the region Q bounded by $z = 3x^2,$ $z = 4 - x^2$, $y = 0$, and $z + y = 6$. [closed]

I'm really struggling to understand and solve this problem, especially when it comes to finding the limit and figuring out how to integrate it.
MAT GG's user avatar
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1 answer
31 views

Integral of Cauchy distribution?

I'm studying for my midterm and got stuck in a basic probability question. The question is as given below. Consider Neyman-Pearson criteria for two Cauchy distributions in one dimension $$p(x|\omega_i)...
JourneyToUngoro's user avatar
-1 votes
0 answers
36 views

How to calculate kink center of mass

I have a kink soliton of $\phi^4$ model: $$ \phi(x,t)=\tanh(x-ct)$$ where $c$ is the velocity, I want to calculate the kink center of mass but I don't know how ? any help please
Espoir's user avatar
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What is the exact asymptotic behavior of $f(x)>0$ such that $\int_c^\infty \frac{1}{f(x)}dx$ converges?

We know that, for positive monotonically increasing functions like $f(x) = \sqrt{x}$ and $f(x) = x$, if you integrate their reciprocals to infinity, it doesn't converge. On the other end, for ...
chausies's user avatar
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-2 votes
1 answer
42 views

Any resources or way I can search things about integrals without the theorem of calculus [closed]

I have so much maths to learn in such a short time and I just can't try to watch another video where all they say is 'well let's think about the antiderivative!' I dont like that to do anything with ...
amnna's user avatar
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1 vote
1 answer
55 views

why does estimation of this proper integral fail?

I have the following integral $ \int_2^\infty \frac{1}{x(\ln x)^p} dx $ and $ \int_0^\frac{1}{2} \frac{1}{x^(\ln x)^p} dx $ I know that if I have something like $ \int_2^\infty \frac{1}{x^q(\ln x)...
nathan's user avatar
  • 23
4 votes
1 answer
41 views

Integral inequality with exact values of a function

$f(x)$ is non-zero continuously differentiable function. Prove that $$\int\limits_1^2 x(f'(x))^2 dx +2f(1)f(2)>0$$ I tried to use Cauchy–Schwarz integral inequality. I also tried integration by ...
perenqi's user avatar
  • 107
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0 answers
17 views

Bound an integral by a function with an appropriate decay

I am reading a result about estimate for the heat equation with right hand side and there is a step of computation I don't quite get. Let $\nu \in (0,1)$, then $$\int_0^t \frac{1}{1 + s^\nu} \frac{1}{...
Falcon's user avatar
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2 votes
3 answers
171 views

Help request for calculating an integral

I am trying to calculate the following inntegral $$\int \frac{2\sqrt{1- x^2}}{2 x\sqrt{1- x^2}+ 5}\, dx. $$ My attempt: $$\int \frac{2\sqrt{1- x^2}}{2 x\sqrt{1- x^2}+ 5}\, dx = \frac{1}{x} - \int \...
Mary_26's user avatar
  • 157
2 votes
1 answer
157 views

Help needed with an integral $\int_0^1 \frac{\ln(x) \ln(1+x) }{1+x^2} dx$.

I am trying to evaluate the integral $$ \int_0^1 \frac{\ln(x) \ln(1+x) }{1+x^2} dx $$ Integration by parts gives \begin{eqnarray*} \int_0^1 \frac{\ln(x) \tan^{-1}(x) }{1+x} dx + \int_0^1 \frac{\ln(x) ...
Donald Splutterwit's user avatar
-4 votes
0 answers
32 views

need help solveing integral [closed]

integral of $\frac{1}{(x+1)\left(\sqrt{1+x^2}\right)}$
Andrew Makarevich's user avatar
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2 answers
60 views

Computation of $\displaystyle\int_0^\infty \frac{e^{-\frac{(x-b)^2}{x}}}{x^k}\text{d}x$

I struggle to compute the integral $I$ defined as follow: $$I := \int_0^\infty \frac{e^{-\frac{(x-b)^2}{x}}}{x^k}\text{d}x$$ with $b>0$ and $k\in\mathbb{N}$. Integral $I$ is well defined First of ...
NancyBoy's user avatar
  • 325
1 vote
1 answer
84 views

Integral of $e^{-x^2}$

As we all know that integral of the function $e^{-x^2}$ is a non elementary function in the form of an error function. But can we calculate it's values at particular points for example the integral ...
Aarush Saharan's user avatar
2 votes
0 answers
22 views

Integral cauchy distriution equals normal distribution?

I'm studying for my midterm and got stuck in a basic probability question. The question is as given below. Consider Neyman-Pearson criteria for two Cauchy distributions in one dimension $$p(x|\omega_i)...
JourneyToUngoro's user avatar
0 votes
0 answers
25 views

Calculating the expected value for a complicated pdf

I am given the following cdf (cumulative distribution function) for the random variable $T$ with support $x\in(0, \theta)$. $$F_{T}(x)=\Bigg(\frac{x-\frac{1}{n}}{\theta}\Bigg)^{n}$$ I am asked to ...
HornyPigeon54's user avatar
1 vote
0 answers
15 views

Question about existence of solutions to integral equations of the first kind

We have three random variables $U, W, A$ and consider the integral operator. The integral operator $T$ is defined as $$Tf= \int f(w,u)p(w|a)dw = p(u|a). $$ for any fixed variable $u$, where $p(w|a)$ ...
叶心萤's user avatar
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2 answers
50 views

Is there a relation between Fubini's theorem and change of variable theorem?

In an exercise, it asks to use the change of variable theorem to calculate a double integral, but then it asks to redo the work using Fubini's theorem. Is there a way to benefit from previous work? In ...
Alia's user avatar
  • 59
6 votes
0 answers
101 views

Evaluate $\int_0^\infty\frac{dx}{1+x^2}\prod_i\arctan a_ix$ (product of arctangents and Lorentzian)

Define $$I(a_1,\dots,a_n)=\int_0^\infty\frac{dx}{1+x^2}\prod_{i=1}^n\arctan a_ix$$ with $a_i>0$. By this answer $\newcommand{Li}{\operatorname{Li}_2}$ $$I(a,b)= \frac\pi4\left(\frac{\pi^2}6 -\Li\...
Parcly Taxel's user avatar
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1 answer
45 views

Why isn't the integral of the total differential of a function equal to the function? [duplicate]

For example let $w = 2x^3y$ then $$dw = \frac{\partial w}{\partial x}dx + \frac{\partial w}{\partial y}dy $$ $$\implies dw = 6x^2y\hspace{1.5mm} dx + 2x^3 \hspace{1.5mm} dy$$ Then why isn't it that $$...
Jack's user avatar
  • 11
3 votes
6 answers
173 views

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

I am trying to show that $$\int_{-\infty}^\infty \frac{x^2}{\left(x^2 + a^2\right)^2} dx = \frac\pi{2a}$$ for $a > 0$ using the Residue Theorem. The formula I am using says $$\int_{-\infty}^\infty ...
Clyde Kertzer's user avatar
2 votes
1 answer
105 views

How to reduce $\int_0^{\pi/2}\frac{1-\sin x}{\cos^2x}\sqrt{\tan x}\,dx$ to complete elliptic integral?

I came across another old post concerning a definite integral whose closed form can be expressed with a complete elliptic integral: $$I = \int_0^\infty \left(\sqrt{1+x^4} - x^2\right) \, dx = \frac1{6\...
user170231's user avatar
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0 votes
1 answer
82 views

What is $\int (y'(x))^2 dx$?

By rearranging $(y'(x))^2)$ to $((\frac{dy}{dx})^2)$ I think I can simplify it to $\frac{(dy)^2}{dx}$. Is that valid? I know that treating derivatives as fractions is not always mathematically valid....
AtomProgrammer's user avatar
3 votes
4 answers
94 views

$\int_0^{2\pi}\arccos\left(\sin\left(x\right)\right)dx$

$$\int_0^{2\pi}\arccos\left(\sin\left(x\right)\right)dx$$ I started by splitting it into multiple intervals $$\int_0^{2\pi}\arccos\left(\sin\left(x\right)\right)dx=\int_0^{\frac{\pi}{2}}\arccos\left(\...
math and physics forever's user avatar
2 votes
0 answers
74 views

A problem that is stumping our whole section in Basic Calculus

We have an assignment that we're supposed to pass by Monday, and all seems to be well until this problem showed up. $\int 9x \sin (x^3)dx$ We're only 2 lessons in to Integrals and have only recently ...
kentisnushki's user avatar
2 votes
1 answer
39 views

An integral inequality with lipschitz function

Suppose that $m=\min_{x\in[a,b]}f(x),K=\int_a^b\frac{dx}{f(x)}$, $f(x)>0$ and it satisfies $\forall x,y\in[a,b],|f(x)-f(y)|\le L|x-y|$. Prove that $\int_a^bf(x)dx\le \frac{e^{2LK}-1}{2L}m^2$. I've ...
TaD's user avatar
  • 133
4 votes
1 answer
64 views

Probability of a random cyclic quadrilateral enclosing a fixed point in its circle

I finally found a single integral solving the natural generalisation of the problem discussed here: For $n\ge1$ pick $n+2$ points uniformly at random on the unit circle. What is the probability $P_n(...
Parcly Taxel's user avatar
1 vote
0 answers
34 views

How to prove that if $\int_{-1}^1 √(1-x^2)f(x) \ dx= \ 0$ for some nonnegtive continuous function $f(x)$, then $f(x)=0$ on $[-1,1]$

How to prove that if $$\int_{-1}^1 √(1-x^2)f(x) \ dx= \ 0$$ for some nonnegtive continuous function $f(x)$, then $f(x)=0$ on $[-1,1]$ Does anyone have any idea about this question? I try to use ...
user1228735's user avatar
2 votes
1 answer
51 views

How to calculate $\int(yy'' + (y')^2)\,dx$?

Context: I have the following second-order differential equation: $$yy'' + (y')^2 = x$$ I noticed that $$yy'' + (y')^2 = (yy')'_x \,\,\,\, (1)$$ So I integrate both sides with respect of $x$: $$\int(...
Rodion Iskhakov's user avatar
1 vote
1 answer
83 views

how to evaluate $\sum_{n=1}^\infty \sum_{m=1}^{n-1} \frac{(-t)^{m+n}}{m-n}$?

I tried to find the summation $$ \Omega=\sum_{n=1}^\infty \sum_{m=1 , m\ne n}^\infty \frac{(-1)^{m+n} H_{m+n}}{m^2-n^2}$$ and got $$ \Omega=\int_0^1 \frac{\ln(1-t)}{t} \left( \frac{t^2}{1-t^2} \ln(1+t)...
Faoler's user avatar
  • 1,321
0 votes
1 answer
39 views

Estimating the parameters of an ellipse (part 3)

This post is a follow up of this and this previous ones. I've found an explanation for the following formulas \begin{equation} \hat{\ell}_1 \triangleq 2\sqrt{\hat{\Lambda}_{11}} \qquad \hat{\ell}_2 \...
matteogost's user avatar
1 vote
1 answer
80 views

Is $\sin x/x$ Lebesgue integrable over $[0,\infty)$?

Let $f(x)=\frac{\sin x}{x}$ if $x\neq 0$ and $f(x)=0$ if $x=0$. Is $f$ Lebesgue integrable? Also, is $\sin^2x/x^2$ Lebesgue integrable? Note that $\lim_{x\to 0}\frac{\sin x}{x}=1=f(0)$. So $f$ is ...
H.Y Duan's user avatar
  • 334
0 votes
1 answer
39 views

Order statistics vs Integration to calculate expectation

I have the following problem - $a$ and $b$ are independently drawn from uniform distribution. $a$ is drawn from uniform distribution $\{m, 1+m\}$ and $b$ is drawn from uniform distribution $\{0,1\}$. ...
Elina Gilbert's user avatar
2 votes
0 answers
34 views

If f is absolutely integrable then f is also integrable?

My real analysis professor said: If $f:\mathbb R\to\mathbb C$ is absolutely Riemann-integrable, i.e. $|f|\in\mathcal R([a,b])$ for any $a<b$, then, by definition, $f$ is Riemann-integrable, i.e. $...
John Davies's user avatar
0 votes
0 answers
27 views

Use of dominated convergence theorem in Manski (1985)

I'm confused by the use of the dominated convergence theorem (DCT) in Lemma 5 of Manski (1985) (see below). Note that $b = (\tilde{b}_1, \dots, \tilde{b}_{K-1}, b_K).$ Specifically: I presume the ...
Giacomo's user avatar
  • 147
0 votes
0 answers
24 views

Prove a function f is Riemann integrable [closed]

I don't know how to answer this question: Let $f:[0,1] \longrightarrow \mathbb{R}$ such that $f(x) = 1$ if $x \in S$ otherwise $f(x) = 0$ if $x \notin S$ where $S=\Big\{ \cfrac{1}{n}: n \in \mathbb{N} ...
Rem's user avatar
  • 43
-1 votes
0 answers
60 views

PKU 2024 spring complex analysis midterm test: integrals [closed]

I'm trying to calculate by residue theorem... The result are allowed to include the Catalan constant. Problems unsolved: (1)$\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{x\sin x}{a+\cos x} \,\mathrm{d}...
wyz's user avatar
  • 1
0 votes
0 answers
62 views

Numerical integration with variation of parameters

I'm having a lot of trouble with a numerical integration problem for an astrodynamics course. We're starting with the function for a perturbed oscillator $$ \ddot{x} + 3 x + x^3 = 0 $$ whose ...
gaharaz's user avatar
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0 answers
20 views

Conservative vector fields: curves and path

Suppose a vector field $F$ has the property that $$ \int_C F =0 $$ for all curves $C$ of class $C^1$. Is it true that the same property holds if we change $C$ for a path (yuxtaposition of $C^1$ curves,...
user210089's user avatar
2 votes
0 answers
81 views

Alternative approach for $\int\frac{1}{\tan\left(\frac{1}{2}\csc^{-1} (x)\right) - \tan\left(\frac{1}{2}\sec^{-1}(x)\right)} \, dx$

This is example 2 in my "Integration Using Some Euler-Like Identities" blog post. $$\int\frac{1}{\tan\left(\frac{1}{2}\csc^{-1} (x)\right) - \tan\left(\frac{1}{2}\sec^{-1}(x)\right)} \, dx\...
Emmanuel José García's user avatar
3 votes
3 answers
112 views

Does the inequality $\iint_{\mathbb R^2} \frac{|f(x)|^2}{|x-y|^a} \ dy\,dx\le k ||f||_{L^2(\mathbb R)}^2$ hold?

I am solving an exercise about integration in dimension 2 and I am stuck at this point. Suppose that $f\in L^2(\mathbb R)$ and let $a\in (1, 2)$. Consider the integral $$\iint_{\mathbb R^2} \frac{|f(x)...
Physics user's user avatar
-1 votes
0 answers
19 views

improper integral in sub interval [closed]

if an improper integral divergence in a specific interval and in that interval the function is non-negative (or non positive) so every interval that contains the specific interval (assuming that the ...
Mohamad Masaada's user avatar
-2 votes
1 answer
70 views

Can $\int_1^3 \sqrt\frac{1+x}x \,dx$ be evaluated using rationalisation? [closed]

Just wondering, is there a way I can evaluate this integral using rationalisation? $$\int_1^3 \sqrt\frac{1+x}x \,dx$$
Jordan Lim's user avatar

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