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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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$I_n=\int_{0}^{1} \arcsin(1-x^n) \,dx $ [closed]

Find the limit of the sequence $(I_n) _n$ wherw $I_n=\int_{0}^{1} \arcsin(1-x^n) \,dx $. I try to integrate by parts but I get $\infty \cdot 0$. I need to practice more examples. Can you recommend ...
ale's user avatar
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4 votes
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Confused by many different definitions of antiderivative

Wikipedia: an antiderivative ... of a function $f$ is a differentiable function $F$ whose derivative is equal to the original function $f$. This can be stated symbolically as $F' = f$. Stewart, ...
user103496's user avatar
-2 votes
0 answers
43 views

Does $\int_1^{\infty } w \sin (w) \, dw$ converge? [duplicate]

Can you help me to show if this integral converges or diverges? $$\int_1^{\infty } w \sin (w) \, dw$$ It oscillates infinitely. Edits: I deleted the plot since the scale was not correct. I also ...
mattTheMathLearner's user avatar
1 vote
1 answer
32 views

Evaluating the inverse trigonometric integral $\int_{0}^{z}\mathrm{d}t\,\frac{2\arctan{\left(at\right)}}{1+t^{2}}$ in terms of polylogarithms

Define the function $\mathcal{J}:\mathbb{R}_{>0}^{2}\rightarrow\mathbb{R}$ via the definite integral $$\mathcal{J}{\left(a,z\right)}:=\int_{0}^{z}\mathrm{d}t\,\frac{2\arctan{\left(at\right)}}{1+t^{...
David H's user avatar
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4 votes
0 answers
50 views

The integral of $\frac{\arctan^2(x)}{x}$ and $\frac{\log(1+x^2) \arctan(x)}{x}$

I was doing some harmonic series and happened to stumble upon the fine result $$\sum_{n=1}^\infty\frac{H_n}{n} x^n=\operatorname{Li}_2(x)+\frac{1}{2}\log^2(1-x), \quad |x|<1$$ I played around with ...
Karabo Dibakoane's user avatar
8 votes
2 answers
111 views

Evaluating $\iint\limits_{0\le x\le y\le1}\!\sqrt{1+x^2-y^2}\,{\rm d}x\,{\rm d}y$

How can I evaluate this integral? $$\iint\limits_{0\le x\le y\le1}\!\sqrt{1+x^2-y^2}\,{\rm d}x\,{\rm d}y$$ I know from Wolfram Alpha that the answer is $\frac13\ln2+\frac16\approx0.39772$. However, ...
Akiva Weinberger's user avatar
0 votes
0 answers
20 views

Integral of sum of two error function squared

Within my research on diffusion through a rectangular domain, I came across this integral $$\int\limits_{-\infty}^{\infty} \left(\textrm{erf}(a+x)+\textrm{erf}(a-x)\right)^2~dx.$$ Can anybody please ...
karim's user avatar
  • 41
4 votes
3 answers
326 views

Integration of derivatives

I've recently finished my high school math finals and have come across this integral that gave me some issues: $$\int f^2(x)f'(x)\,dx$$ (the exact function does not matter here) Initially, I solved ...
DoubleYouSlash's user avatar
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0 answers
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Evaluate $\int_{0}^\infty \frac{x^{a-1} \ln(x)}{1+x^2} dx$ where: $0<a<1$. [closed]

$\int_{0}^\infty \frac{x^{a-1} \ln(x)}{1+x^2} dx$ where: $0<a<1$. I understand that it requires series expansion but thats about as far as I've gotten, really stuck from the get-go. ...
keixx's user avatar
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2 votes
2 answers
105 views

Computing $\int_0^1 \frac{x f(x)}{1+x^2} d x$, where $f(x)$ are the inverse of trigonometric functions.

Recently, I started to investigate the integrals involving the inverse of the six trigonometric functions $f(x)$: $$\int_0^1 \frac{x f(x)}{1+x^2} d x$$ by integration by parts as $$ \begin{aligned} \...
Lai's user avatar
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0 votes
2 answers
51 views

Letter under double and triple integrals [closed]

There are several letters that can be under a double integral, like $S$, $R$, or $C$ for line integrals. What do all of these letters stand for? $$ \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \...
ddodd69's user avatar
1 vote
1 answer
96 views

What does $\mathrm{d}(x^2)$ mean in the $\int_a^bf(x)\, \mathrm{d}(x^2)$ or is such a thing not valid

Here is my understanding of the $\int_a^bf(x)\, \mathrm{d}x$: \begin{align} \int_a^bf(x)\, \mathrm{d}x = \lim_{n \to \infty} \frac{b - a}{n} [f(a) + f(a + h) +... + f(a + (n-1)h)] \end{align} where $h ...
ADITYA VIKRAM SINGH's user avatar
4 votes
4 answers
164 views

Solving $\int_0^1(\ln x)^{2024}dx$ [duplicate]

Evaluate $$I=\int_0^1(\ln x)^{2024}dx$$ I solved this by finding a pattern with "Integration By Parts" but ran into a few issues with the lower bound being $0$. After doing "Integration ...
Grey's user avatar
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-3 votes
1 answer
66 views

Proof of the General Change of Variables Theorem in Rn?

I was reading the theroems on site https://planetmath.org/changeofvariablesinintegralonmathbbrn and found the following general change of variables theorem: Theorem 2. Let g:X→Rn be continuously ...
Patrick Oliveira's user avatar
7 votes
1 answer
1k views

Conceptual Issue with Deriving Sphere Volume Formula [closed]

I am trying to derive the sphere/hemisphere volume formula by using the same concept (polar coordinate) when deriving the surface area: SURFACE VOLUME The math looks ok but I cannot explain to my ...
Math_Physics's user avatar
1 vote
1 answer
107 views

$I_n=\int_{0}^{\pi} e^{-n \sin x}\,dx $ [duplicate]

Study the convergence of the sequence $$I_n=\int_{0}^{\pi} e^{-n \sin x}\,dx $$ and find its limit. My idea: $I_n= \int_{0}^{\frac{\pi}{2}} e^{-n \sin x}\,dx + \int_{\frac{\pi}{2}}^{\pi} e^{-n \sin x}\...
ale's user avatar
  • 1,768
2 votes
0 answers
67 views

how to find closed form for ${}_3F_2(-1)$ function identity?

I tried to prove this identity I got it from wolfram functions site $$\Omega_a={}_3F_2(1,1,a;2,3-a;-1)=\frac{2-a}{2(1-a)} H_{1-a} $$ firstly from its series I got that $$ \Omega_a=\frac{\Gamma(3-a)}{\...
Faoler's user avatar
  • 1,657
6 votes
1 answer
216 views

How to evaluate the integral $\int_0^{\infty} \frac{\sin(k \ln(t))}{\sqrt{t}} \left\{\frac{1}{t}\right\} \, dt $

I am trying to evaluate the integral $$ \int_{0}^{\infty}\frac{\sin\left(k\ln\left(t\right)\right)}{\sqrt{t}}\, \left\{\frac{1}{t}\right\} \, dt, $$ where $\left\{\...
stephan's user avatar
  • 437
-1 votes
0 answers
54 views

How to compute Contour Integral Numerically [closed]

So I am trying to compute the integral from $0$ to $a$ on the real axis, shown in the picture, for a function that is completely analytic on the upper half plane except at $E + i\epsilon$ where $\...
Aziz's user avatar
  • 9
0 votes
1 answer
48 views

How do I approach this volume problem?

I'm trying to find the volume of a body bounded by the surface given by the equation $(x^2 + y^2 + z^2)^2 = (x^2 - y^2)z$, where $z \geqslant 0$. Here's my thought process: by observing that the we ...
sudstepan's user avatar
1 vote
0 answers
20 views

Excess part from integration of graph

Photo of Graph of a arbitrary function on which integration is performed Here, in the photo, using the basic idea of integration, we make a long rectangular block to cover the area under the graph. ...
Akhilesh G's user avatar
1 vote
1 answer
39 views

Double integral of $\iint x\,\mathrm dx\,\mathrm dy$ using polar coordinates

Solve$\newcommand{\dd}{\mathrm d}$ $$\iint_D x\,\dd x\,\dd y$$ using $x= 2r \cos \phi, y= r \sin \phi$ over D: $x^2+4y^2 \leq 4, x \geq 0$. I obtain $r \leq 1$ and $\frac{3\pi}{2} \leq \phi \leq \frac{...
TerribleStudent's user avatar
-1 votes
1 answer
55 views

Notation for double integral w.r.t. same variable [closed]

When taking the double integral of a function in terms of the same variable $x$, should I write $\int\int y~\text d^2x$ or $\int\int y~\text dx^2$, and why?
GPWR's user avatar
  • 222
-1 votes
0 answers
34 views

Find a recurrence relation for a polynomial integral [duplicate]

Its expression is: $$\int_0^1{(x-x^2)^n}, dx$$
liverpoolever's user avatar
1 vote
1 answer
93 views

How to Solve $\int \frac{x + 1}{(x^2 + 6x + 14)^3} dx$? [duplicate]

I am trying to solve the following integral and would appreciate some guidance: $$\int \frac{x + 1}{(x^2 + 6x + 14)^3} \, dx$$ I have attempted various methods, including substitution and integration ...
Math124's user avatar
  • 37
1 vote
2 answers
82 views

Complex integrals that look like they agree, differ by sign (according to Mathematica)

Consider the integral $$\int_0^\infty \frac{dz}{1-z^2 +i0^+},$$ I would assume it to agree with the integral $$\int_0^\infty \frac{dz}{(1-z+i0^+)(1+z+i0^+)}. $$ However, according to Mathematica the ...
VerwirrterStudent's user avatar
0 votes
0 answers
70 views

Double integral $ \iint_D (x^4-y^4) dx\,dy$

I have troubles with the following integral $$ \iint_D (x^4-y^4) dx\,dy $$ over D: $1<x^2-y^2<4, \sqrt{17}<x^2+y^2<5, x<0, y>0$ This is the same problem as in Compute $\iint_D (x^4-y^...
TerribleStudent's user avatar
-3 votes
0 answers
46 views

Arc length derivation [closed]

enter image description here Why does the hypothetical X value Xi that would equal the avg slope of a section get replaced by X? I don't remember using the mean value theorem in the definition of the ...
Brayden Mooney's user avatar
0 votes
0 answers
37 views

Moments of Pearcy type integral

In my research I encounter Moments of Pearcy Integral which can be written as $$ \int_{-\infty}^{\infty}x^{n} {\rm e}^{-ax^{4} + bx^{2} + cx}\,{\rm d}x\qquad a > ...
Sam Hilary's user avatar
2 votes
1 answer
99 views

Constant term in the Euler-Maclaurin expansion of $s_n=\sum_{k=1}^n \tfrac{1}{k+1/2}$

This question is a followup from my previous post based on the Euler-Maclaurin formula: How to find the correct constant term with Euler-Maclaurin formula, $\sum_{j=1}^n j\log j$. This time I am ...
bob's user avatar
  • 2,249
0 votes
0 answers
51 views

Approximation of a Riemann sum.

Given a twice continuously differentiable function $f\in C^2([0,1])$, is there a theorem/result/algorithm on how to place $0<x_1<\ldots<x_{n-1}<1$ so that adding $x_0=0$ and $x_n=1$, $$ \...
username's user avatar
  • 208
-1 votes
2 answers
65 views

$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$.Calculate $\int_A xyz \ d \lambda_3$. I need to verify my solution.

$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$ Calculate: $$\int_A xyz \ d \lambda_3$$ Solution: We know that: $x^2 + y^2 + z^2 > 0$ and therefore $2x + 2y > 0 \iff x + y > 0$ ...
thefool's user avatar
  • 1,118
0 votes
0 answers
47 views

Conditions for function to be periodic

I am investigating the following type of functions \begin{equation} I(\alpha) = \int_{0}^{\pi}f(t)\cos(\alpha t)\,\mathrm{d}t\,. \end{equation} where $f(t)$ is a real-valued function with non-negative ...
Dennis Marx's user avatar
0 votes
0 answers
26 views

Will the following Method of engineering analysis work?

Analytical Engineering Analysis of 3D Shapes Using volume integral($\iiint_{}^{}{f(t)}dx dy dz$) to do a Analytical Engineering Analysis of 3D Shapes without using mesh based FEA. Like integration ...
Diji Jayakaran's user avatar
19 votes
9 answers
2k views

Formula for bump function [duplicate]

I would like to formulate a bump function (link) $f(x)$ with the following properties on the reals: $$ f(x) := \begin{cases} 0, & \mbox{if } x \le -1 \\ 1, & \mbox{if } x = 0 \\ 0, & \...
Richard Burke-Ward's user avatar
1 vote
2 answers
38 views

Laplace transform of $\sin(\omega t)$

I am learning about the Laplace transform and I know I got the answer to this example question wrong, but I'm trying to figure out if I just made a calculus or algebra type error, or if I'm ...
Keara's user avatar
  • 125
-2 votes
0 answers
47 views

Euler Substitution same question 2 cases [closed]

I have been trying to figure out the integral of: here we can use Euler substitution 1 and 2: 1st one for a(1) > 0 where sqrt(x^2-x+1) = t-x and 2nd for c(1) > 0 where sqrt(x^2-x+1) = tx-1.. ...
ANNJEE's user avatar
  • 49
2 votes
1 answer
93 views

M is composed of line segments connecting ellipse to $(0,0,0)$ Calculate integral $\int_M \sqrt{x + 3z}\ d \lambda_2$ over those. Almost done.

I found such an exercise among my set of exercises preparing for exams and I have no idea how to solve that. Every point of ellipse $\{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1, x + z = 1 \}$ is ...
thefool's user avatar
  • 1,118
0 votes
1 answer
64 views

Help Solving $\int\text{erf}(x)\cos(\alpha x)dx$

I'm trying to solve: $$\int\text{erf}(x)\cos(\alpha x)dx$$ where $\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}dt$ I am solving this as part of my attempt to find the Fourier series for $\text{...
Grey's user avatar
  • 805
1 vote
0 answers
40 views

Simplification of Dependent Nested Integrals

I have been working with nested Integrals of the following form : $$ \int_0^{T}dt_n \int_0^{t_n}dt_{n-1} \cdot \cdot \cdot \int_0^{t_2} dt_1 \prod_{i=1}^n f_i(t_i) $$ Some simpler cases and prework : ...
kodel's user avatar
  • 11
1 vote
2 answers
134 views

Find $\displaystyle\lim_{n \to \infty} \int_0^\infty \frac{1+\frac{x}{\sqrt{n}}e^{-x/n}}{(x+1)^2} \, dx$

Find $\displaystyle\lim_{n \to \infty} \int_0^\infty \frac{1+\frac{x}{\sqrt{n}}e^{-x/n}}{(x+1)^2} \, dx$. I have tried a couple methods. As a first method, it is tempting to bound the integrand above ...
Grigor Hakobyan's user avatar
0 votes
0 answers
32 views

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

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
0 votes
1 answer
38 views

Find a function which is not absolutely integrable but is the sum of absolutely integrable functions

The exercise asks me to show an example of functions $f,g: [a,b] \to \mathbb{R}$, such that $ |f(x)|, |g(x)|$ are integrable on the interval $[a,b]$, but $|f(x)+g(x)|$ is not integrable on that ...
khalelbm's user avatar
7 votes
3 answers
905 views

This expected value has a minimum!

Problem. Let $X$ be a positive, real random variable whose probability density function is bounded by $1$. Prove that $E[X]\geq \frac 12$. Hi everyone. This problem is essentially saying that the ...
aleph2's user avatar
  • 986
2 votes
1 answer
229 views

How to evaluate $\int_0^{\infty } \frac{\sin (\pi x)}{\log (x)} \, dx$

I have tried different substitutions and transformations, but am not getting a lead. Any suggestion would be helpful. The numerical value of the integral is around $-3.2192$. Interestingly these two ...
mattTheMathLearner's user avatar
0 votes
2 answers
129 views

Variable transformation in the definite integral

recently I encounter a variable transformation problem in the derivation and I did not figure out how it works. $$\int_0^1\int_0^1\frac{\partial^2}{\partial\rho^2}\{s\rho^2C(\textbf{r}_2,ss^\prime\rho)...
Xeh Deng's user avatar
0 votes
0 answers
37 views

Approximation or an analytical form of integrals

I'm working on a bayesian rule of wavelet shrinkage using the raised cosine distribution as a prior distribution for $\theta$. Its expression is given by: $$\delta(d) = \frac{(1-\alpha) \int_{\frac{-\...
juliana marchesi's user avatar
-5 votes
0 answers
55 views

A simple yet complex proof that I am unable to solve. [closed]

prove that: ((e^(ax ))cos(bx))^n=((sqrt(a^2+b^2))^n)(e^(ax))cos(bx + n.arctan(b/a))
Rajrup Chattopadhyay's user avatar
0 votes
0 answers
79 views

Looking for citation of a definition of multivariable integral

I need a definition of Multivariable Riemann Integral to cite in my article. I've been searching different cited sources, for example Rudin, Folland, Stewart and Larson books. In the first two I ...
Maiia Bakhova's user avatar
0 votes
0 answers
49 views

Am I correctly applying repeated integration by parts?

Say we have two compactly supported functions $f,g:\mathbb{R}^n\to\mathbb{R}$. I found myself computing \begin{equation} \begin{split} \int_{\mathbb{R}^n}f\frac{\partial^{r}g}{\partial x^{\alpha_1}\...
Sam's user avatar
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