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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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Need help with the steps and limits in this multivariable integration of a joint probability density function

I am not sure how to proceed with this double integration. I know this can be evaluated in a much easier way than solving the integral as its just the volume of a cube but I need help with the process ...
Kumar Yashasvi's user avatar
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0 answers
28 views

Intergrating $ \int_{M} f (x.,y,z,w)\ d {\rm Vol}_3 $

I want to integrate $$ I = \int_{M} f (x.,y,z,w)\ d {\rm Vol}_3 $$where $f(x,y,z,w) = (x+y)e^{z+w} $, and $M = \{x+y+z+w = 1, x,y,z,w > 0\} $. I need to find a parameterization of M; if I consider $...
FNB's user avatar
  • 391
-1 votes
0 answers
38 views

The moment of multivariate normal distribution

This is a computational problem I ran into while reading an article. I describe my question below: Let $\boldsymbol{Z}\sim N(0,I_{p\times p})$ and $\boldsymbol{y}_{i}\in \mathbb{R}^{p}$. We need to ...
Lop's user avatar
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-2 votes
1 answer
39 views

If $f$ and $g$ coincide almost everywhere on $[a, b]$, then is $\int_a^b f(x) dx = \int_a^b g(x) dx$? [duplicate]

Let $a$ and $b$ be any real numbers such that $a < b$, and let $S$ be a (nonempty) subset of the closed bounded interval $[a, b]$ such that $S$ has measure $0$. Now let $f \colon [a, b] \...
Saaqib Mahmood's user avatar
1 vote
1 answer
36 views

What is the fault in this method of finding second moment of area of a circle

I am trying to find the second moment of area of a circle about a diameter using first principles. Place the centre of the circle at the origin of XY-plane. Now consider a tiny circular sector with an ...
Jarvis's user avatar
  • 139
0 votes
0 answers
15 views

Proof of Non-Exactness for Polynomials of Degree p in Quadrature Formulas of Order p

Given a Quadrature Rule $$ \int_{a}^{b} f(x)dx \approx (b-a) \sum_{k=1}^s b_k f(a+c_k(b-a))$$ of order $p$ $$\frac{1}{q} = \sum_{k=1}^s b_k^{q-1} for\:all\: q=1,...,p\:, but\:not\:q=p+1$$ Show that ...
trsommer's user avatar
  • 117
-1 votes
0 answers
95 views

$f(x) = k(x-a)(x-b)(x-c)(x-d)(x-e)(x-t)$ [closed]

Question - $f(x) = k(x-a)(x-b)(x-c)(x-d)(x-e)(x-t)$. $f^{\prime}(x) = 6(x^5-8x^4+24x^3+Ax^2+Bx+c)$ if $ e\leq k_1 $ and $t \geq k_2$. $a<b<c<d<e<t$. Find $k_1+k_2$. My attempt - I ...
User_X's user avatar
  • 11
3 votes
3 answers
193 views

Question regarding integral involving logarithm and sine

I have to compute the following integral $$\int_{0}^{\pi/2} \frac{\ln(1-\sin x)}{\sin x} dx$$ I decided to solve this using the Feynman's Trick for integration and parametrized the integral as follows ...
koiboi's user avatar
  • 358
-4 votes
0 answers
35 views

Calculate the Following Complex Integral Around the Given Contours [closed]

$$ \oint_{\lambda} \frac{\cos^5(z)}{(z - i\pi)^3} \, dz $$ $$ \|z\| = 1 \\ $$ $$ \|z + 1\| = \frac{\pi}2 $$ I have a question regarding the calculation of the integral. When using the residue theorem ...
Armando Cruz's user avatar
4 votes
1 answer
56 views

Integrate the product of a heaviside step and the absolute value?

I have a rather tricky integral here: $$\underbrace{\int_0^R r_0\Theta(R-r_0)|r-r_0|dr_0}_{(1)} - \underbrace{\int_0^R r_0\Theta(R-r_0)|r+r_0|dr_0}_{(2)} \ \ \ \ \cases{0\le r < \infty \\ R=1}$$ ...
Researcher R's user avatar
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0 answers
24 views

Solving challenging 4D integrals arising from triangle-triangle gravitational interaction

I am trying to find a closed form for two related integrals, coming from a physics problem partially solved here, about attractive forces between two triangles : $$\begin{align} {\bf F}_1 &= -G ...
user1420303's user avatar
3 votes
0 answers
34 views

Determining the significance of a curve's factors

Given the equation $x^2+x+1$ you could easily determine that $x^2$ will have the greatest overall impact on the curve--then $x$ and finally $1$. And this holds true for any coefficients present as the ...
SlavaCat's user avatar
1 vote
0 answers
55 views

How to formally describe the illustrated area?

I am wondering how I could formally but concisely describe the sum of blue and red areas on the plot below for a cubic spline interpolation (red dashed line) with knots at the red dot points. This sum ...
ufghd34's user avatar
  • 91
3 votes
1 answer
175 views

Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$

While exploring possible applications for exponential substitution, I stumbled upon the following integral identity: $$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
Emmanuel José García's user avatar
1 vote
0 answers
34 views

Mass conservation for a transport equation in mild form

Let us consider the following partial differential equation for $f = f(x,v,t),$ with $x \in \mathbb{T}^1, v \in \mathbb{R}$ and $t \in [0, \tau]$: $$ \partial_t f(x,v,t) + v \cdot \partial_x f (x,v,t) ...
kumquat's user avatar
  • 169
4 votes
2 answers
126 views

Reference for $\int_{-\infty}^{\infty}e^{a x^4+b x^3+cx^2}dx\;$?

In my research I encounter an integral of the form $$ \int_{-\infty}^{\infty}{\rm e}^{\large ax^{4}\ +\ bx^{3}\ +\ cx^{2}}\,{\rm d}x\qquad a < 0,\quad b, c \in \mathbb{R} $$ So the integral is ...
Sam Hilary's user avatar
-1 votes
1 answer
37 views

Recursive piecewise integral formula

I have the recursive formula for the integral $1/(x^2+a^2)^n$, which is, in fact, the one that Ng Chung Tak provides in this link. My problem is that when finding a specific integral, for the case $n=...
Emerson Villafuerte's user avatar
-2 votes
2 answers
146 views

What is the sign of $I_n = \int _{0}^{1}\frac{x^{2n+1}}{x^{2}+1}dx$ [closed]

I was given an exercice to calculate $I_0$ and then $I_0 + I_1$ and then deduce $I_1$, and then asks the sign of $I_n$, can someone help? I tried deductive reasoning but I don't know how to complete ...
Nassim Ouali's user avatar
1 vote
0 answers
53 views

Differentiation under integral signs as done in basic quantum mechanics

In various text books, lectures or lecture notes on basic quantum mechanics, I've seen cases differentiating under integral signs and I am wondering why it is allowed in those situations. The typical ...
russoo's user avatar
  • 2,437
0 votes
1 answer
61 views

Electric field in the plane of a charged ring

It's basically this question Electric field off axis inside a charged ring., but I want to know if it is possible to solve this integral analitically. \begin{align} E= \dfrac{k\lambda}{R}\underbrace{\...
AWanderingMind's user avatar
0 votes
0 answers
25 views

Change of variables Gamma and Dirichlet distribution

I am struggling to solve the following problem: Let $(D_1, D_2, ..., D_n) \sim \text{Dir}(\alpha_1, \alpha_2, ..., \alpha_n), \quad \alpha_1, ..., \alpha_n > 0$ and $G \sim \text{Gamma}(\sum\...
user675763's user avatar
0 votes
0 answers
43 views

Prove/disprove upper bound and lower bound of the Integral

Hey I need to Prove or disprove this sentence: $$ \frac{4}{9}(e-1) \leq \int_0^1 \frac{e^x}{(1+x)(2-x)} \, dx \leq \frac{1}{2}(e-1) $$ using the infimum and supremum method for integrals, where m and ...
miiky123's user avatar
0 votes
0 answers
63 views

Landau Notation Problem

I have this function $$ K_{n} = \int_{1}^{+\infty}\frac{1}{(1+t^2)^n}dt$$ $$ \text{Let }t\geq1,t^2+1\geq1+t\Leftrightarrow\frac{1}{1+t^2}\leq\frac{1}{1+t} \text{ and for } n \in {\mathbb{N^{*}}} : \...
diplodocass's user avatar
0 votes
2 answers
59 views

Integration of function $ [ \int_{0}^{\pi} |\sin x - \cos x| \, dx ] $ [closed]

Hey I need to evaluate this definite integral: $ [ \int_{0}^{\pi} |\sin x - \cos x| \, dx ] $ Don't really know how to approach this, would glad if someone can show me the way to solve this. I can't ...
miiky123's user avatar
0 votes
0 answers
29 views

Related to double integration

In my research work, I got the following expression: $\int_0^{\infty} \int_0^{Z_{lim}} 1-\exp(-\lambda*\frac{(\gamma_{th}([A+\beta^2y^2](P_u/jj1)+1/jj1)-\beta^2yz(\alpha_u-\gamma_{th}\alpha_t))}{\...
Heretolearn's user avatar
4 votes
0 answers
73 views

Validity of Python-derived solution for contour integral $\oint f(z)f(z-\overline{z})~dz$

$\newcommand{\on}[1]{\operatorname{#1}}$ $$ \mbox{Consider the function:}\quad \on{f}\left(z\right) = \frac{{\rm e}^{tz}}{\left(1 + z^{2}\right)^{3}}\, \left(\sqrt{t} - t\right)\ \ni\ t,z \in \mathbb{...
MASTER DHRUV's user avatar
0 votes
0 answers
24 views

Integral of Poisson Kernel

This doubt comes from Dupaigne's book named stable solutions of elliptic partial differential equations. The Poisson Kernel is \begin{equation} P(x,y)=\frac{\partial G(x,y)}{\partial n_{y}}=\frac{1-|x|...
Richard's user avatar
  • 89
-2 votes
0 answers
94 views

Evaluating $\int_{-\pi/2}^{\pi/2} \frac{\sin^2(nx)}{\sin^2(x)(1+e^x)}\,dx $ [closed]

Try this amazing integral....This will be either too much pain or too much fun $$ \int_{-\pi/2}^{\pi/2}\frac{\sin^{2}\left(nx\right)}{\sin^{2}\left(x\right)\left(1 + {\rm e}^{x}\right)}\,{\rm d}x $$
X_xBABAIx_X's user avatar
0 votes
0 answers
70 views

Evaluating $\int_t^Te^{-a_r(T-x)}\cdot\sqrt{s^2(f(a_n))^2+n^2(f(b_n))^2+2snp\,f(a_n)f(b_n)\;}dx$, where $f(k)=\frac1k(1-e^{-k(T-x)})$

Hello:) i am currently working on the Jarrow Yildirim model with a Two factor Hull & White process. In order to derive a expected value i have to calculate this deterministic integral: \begin{...
Valentin's user avatar
0 votes
0 answers
38 views

Solving integral involving minimum $\int_{L^n} \bigwedge_{j=1}^n \Big( \sum_{k=1}^j x_k \Big) \ dx$.

Consider $L = [s,t]$ with $s<t$ and write $a \wedge b := \text{min}(a,b)$. I am trying to solve the following integral. $$ \int_{L^n} \bigwedge_{j=1}^n \Big( \sum_{k=1}^j x_k \Big) \ dx $$ My ...
justAGuy's user avatar
2 votes
0 answers
119 views

Finding a general expression for the improper integral $\int_0^\infty K_1( ( k^2+\alpha^2)^{1/2}r)\sin(kz)\,\mathrm{d}k$

$\newcommand{\on}[1]{\operatorname{#1}}$ In solving a classical fluid mechanics problem involving flow in porous media, I encountered a delicate infinite integral ...
Siegfriedenberghofen's user avatar
-1 votes
0 answers
35 views

Cauchy Schwarz inequality with modulo

I can't understand why integral $\int_x^y{(y-t)dt}$ became $|\delta x|$. By Cauchy Schwarz inequality $$|\int_x^y{(y-t)f''(t)dt}| \leq (\int_x^y{(y-t)^2dt})^{0.5}(\int_x^y{f''(t)^2dt})^{0.5}$$ ...
Pyrettt Pyrettt's user avatar
-4 votes
1 answer
94 views

I really need verification and validation of my view of integrals! [closed]

Questions: A runner is running at a rate of $3 \frac{m}{s}$. How long has he traveled in $4$ seconds? A runner is running at a rate of $3x \frac{m}{s}$. Thoughts: For question one, given that the ...
Idiamine's user avatar
1 vote
0 answers
15 views

Boundary terms of integration by part depends on the order to exchange the differetial operator

I have a very simple question while learning calculus. Suppose $\Omega\subset\mathbb{R}^2$ is a smooth domain. $f,g\in C^\infty(\Omega)$. We consider the integration by part here: $$\begin{aligned} \...
Holden Lyu's user avatar
5 votes
3 answers
114 views

Question About the Antiderivative of $x \sin(\frac{1}{x})$

Since the function $f(x) := x \cdot \sin(\frac{1}{x})$ is continuous everywhere, there is an antiderivative $F(x)$. This should theoretically be continuous, as it must be differentiable. However, if I ...
Noctis's user avatar
  • 236
0 votes
0 answers
68 views

Theorem 7.48 in Apostol's MATHEMATICAL ANALYSIS, 2nd ed: Lebesgue's Criterion for Riemann Integrability [closed]

Here is Theorem 7.48 (Lebesgue's Criterion for Riemann Integrability) in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition: Let $f$ be defined and ...
Saaqib Mahmood's user avatar
2 votes
1 answer
43 views

Integral of $\int_{\Re\lambda=\gamma_0} \frac{e^{\lambda t}}{\lambda^2+c^2}d\lambda$ where $\gamma_0>0$ and $c>0$.

I want to deform the contour to the imaginary axis but avoiding the poles $\lambda=\pm ic$. To avoid the poles one can use half-circles $C_\pm=\{\lambda\in\mathbb C:\lambda = \pm ic+\epsilon e^{i\...
schrodingerscat's user avatar
-2 votes
1 answer
98 views

Is it possible that : $\int_{a}^{b}e^{\ln(x)}dx=\int_{a}^{b}xdx$? [closed]

Is it possible that: $\int_{a}^{b}e^{\ln(x)}dx=\int_{a}^{b}xdx$? Suddenly this question came in my mind after learning about definite integrals in detail from the book written by K.C. Sinha. I ...
Deb Subha Deepa's user avatar
-2 votes
1 answer
61 views

If the integral of a monotonic f converges, does it mean f approaches 0? [closed]

I have come across this question: Say $f(x): [0,\infty) \rightarrow \mathbb{R}$ is monotonic non-increasing, and $\int_{0}^{\infty} f(x)dx$ converges. Does it mean that $\lim_{x\to\infty}{f(x)}=0$? If ...
Nadav Menirav's user avatar
-1 votes
0 answers
51 views

Find the number of solution for an equation $f(x)=0$ we have $f(x)=3x^2+2ax+b\;$ and $\int_{-1}^{1}|f(x)|dx<2$ [closed]

We consider the function $f$ defined by $f(x)=3x^2+2ax+b\;$; $a$ and $b$ are two reals numbers. we have : $$\int_{-1}^{1}|f(x)|dx<2$$ The numbre of solution of the equation $f(x)=0$. chose the ...
user579102's user avatar
0 votes
0 answers
36 views

How can volume be calculated if one parameter is undefined because another goes to zero.

I have the following equations: $$V = \int^{2\pi}_0\int^{r_i}_0hrdrd\phi$$ $$hr = \frac{R^2\sin^3\beta}{(1-\cos\beta\cos\phi)^2}$$ and $$r_i = \frac{R\sin\beta}{1- \cos\beta\cos\phi}$$ When the double ...
rdemo's user avatar
  • 341
2 votes
0 answers
71 views

Abel-Plana vs Euler-MacLaurin summation for $S = \sum\limits_{k=1}^{\infty}\left[ 2 \pi k - 2 - 4k \tan^{-1}(2k)\right]$

I have a sum as follows: $$S = \sum\limits_{k=1}^{\infty} 2 \pi k - 2 - 4k \tan^{-1}(2k) \approx -0.250854$$ $$S=2+\sum\limits_{k=0}^{\infty} 2 \pi k - 2 - 4k \tan^{-1}(2k)$$ Applying Euler-MacLaurin ...
Srini's user avatar
  • 844
0 votes
1 answer
33 views

Showing integrability of f+g and additivity of the Darboux integral

I am currently working on the following question from Measure, Integration & Real Analysis by Sheldon Axler: Suppose $f,g:[a,b]\to\mathbb{R}$ are Riemann (Darboux) integrable on $[a,b]$. Prove ...
Alice's user avatar
  • 508
-1 votes
0 answers
68 views

Need Help Simplifying a Series Involving Exponential and Factorial Terms

I'm working on solving a fractional differential equation and encountered the following series: $$ \sum_{n = 1}^{\infty} \frac{\displaystyle\beta^{n}n^{p}\,{\rm e}^{...
Sujeethan's user avatar
-3 votes
0 answers
51 views

Fourier transform of exp(-1/x^2) [closed]

can someone please show the derivation of Fourier transform of exp(-1/x^2) if it exists?
user1229009's user avatar
5 votes
0 answers
60 views

Integrating the Beta Function

As a learning exercise, I am trying to find the mean and variance of the Beta Probability Distribution (https://en.wikipedia.org/wiki/Beta_distribution) from first principles (i.e. Method Of Moments): ...
wulasa's user avatar
  • 399
1 vote
2 answers
136 views

Integral of ln(sin(x)) [duplicate]

So I've had a few attempts at this this integral, but they all seem to get different answers to wolframalpha. This is my working out: $I=\int \ln(\sin(x))dx \\ I=\int \ln(\frac{e^{ix}-e^{-ix}}{2i})dx \...
The_Lord_ 26's user avatar
-2 votes
1 answer
75 views

triple integral over a complicated region [closed]

$$ \mbox{Can anyone help me evaluate this integral ?:}\quad \iiint_{B}\left(y - x\right){\rm d}x\,{\rm d}y\,{\rm d}z $$ where \begin{align*} B & := \left\{\left(x,y,z\right) \in \mathbb{R}^{3} \...
joaodesousaluz's user avatar
0 votes
0 answers
48 views

Substituting $u=\sin(\theta)$ into $\int_{0}^{\pi} \cos\left(\sqrt{1-\sin^2(\theta)}\right) d\theta$ [duplicate]

I would like to show that $\int_{0}^{\pi} \cos\left(\sqrt{1-\sin^2(\theta)}\right) d\theta = \int_{-1}^{1} \frac{\cos\left(\sqrt{1-u^2}\right)}{\sqrt{1-u^2}} du$ by substituting $u=\sin(\theta)$. ...
Cristof012's user avatar
0 votes
0 answers
41 views

Leibniz rule when integrand has discontinuity

For $0<y<1$ consider, $$F(y)=\int_0^y \frac{-\log^3 x}{1-x} dx$$ I need to prove that $$F'(y)=\frac{-\log^3 y}{1-y}, 0<y<1$$ By definition $$F'(y)=\lim_{h\to 0}\frac{F(y+h)-F(y)}{h}$$ So ...
Max's user avatar
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