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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Intergral of Hypergeometric series

Is it possible to calculate this integral ? $$I:=\int_0^{\gamma^2} \int_0^{\gamma^2} \left((1-x)(1-y)\right)^{s-2} {}_3F_2\left(s,s,s;1,1;xyt\right) dx dy$$ where $\gamma,\; t,\; s\geq 0$ and ${}...
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2 votes
1 answer
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Evaluating $\int_{-\infty}^{\infty}\frac{\ln\left(1+x^{8}\right)}{x^{2}\left(1+x^{2}\right)^{2}}dx$

(Motivation) Here is an integral I made up for fun: $$\int_{-\infty}^{\infty}\frac{\ln\left(1+x^{8}\right)}{x^{2}\left(1+x^{2}\right)^{2}}dx.$$ WolframAlpha doesn't seem to come up with a closed form, ...
  • 2,854
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1 answer
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Multivariate definite integral of an exponential of a quadratic polynomial

I am trying to evaluate the definite integral below where $a, b, c$ are real: $$ \int_{-1}^1 \int_{-1}^1 e^{-(ax^2 + bxy + bx^2)} ~ \mathrm{d} x ~ \mathrm{d} y $$ I know that this can be done easily ...
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1 answer
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Help understanding integral substitution

I don't quite understand the steps involved in this indefinite integral with respect to $t$: $$\int\frac1{M}\frac{dM}{dt}$$ The explanation I've does this substitution: $$u = M(t)$$ Which seems to ...
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1 vote
1 answer
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How to fine the formula for the volume using cylindrical shells?

Use cylindrical shells to find the volume of the solid. The region bounded $y = x^2$ and $y = 1$ is revolved around the line $y = 3$. The answer is: $$\int_0^1 4\pi(3-y)y^{1/2}dy$$ Why is it $4\pi$? I ...
-5 votes
0 answers
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Geometrically Argue that a>0

Please Help: I am having issues with question b and c
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Are there an infinite number of ways to divide the area under the curve?

I came across the trapezium rule and started wondering: if definite integration can be defined as an attempt to approach an infinitely fine approximation of the area under the curve, can the area ...
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Why do we need Jacobian for change of variables in integration? Why do not we change variables in a straightforward way? [duplicate]

I know that Jacobian is necessary to change the area element in integration, for example, \begin{equation} \Delta A= dx dy= \left|\frac{\partial x,\partial y}{\partial r ,\partial \theta}\right| drd\...
3 votes
0 answers
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$\int_{0}^{1}\ln^2\Gamma(x)\ln\Gamma(1-x)\ dx$

I came across the following integral $$\int_{0}^{1}\ln^2\Gamma(x)\ln\Gamma(1-x)\ dx$$ and I have no idea where to even start. I know you can interchange the $x$ and $1-x$ by king's property, and that ...
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2 answers
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$\int_{0}^{\pi/2}\ln^3(\sin x)\,\mathrm dx$

There are closed forms for $\int_{0}^{\pi/2}\ln(\sin x)\,\mathrm dx\,$ and $\,\int_{0}^{\pi/2}\ln^2(\sin x)\,\mathrm dx\,$ but I can’t seem to find a closed form for $$\int_{0}^{\pi/2}\ln^3(\sin x)\,\...
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1 answer
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Are the usual polar coordinates actually diffeomorphisms?

Lets have look at the usual polar coordinates in two dimensions, given by the mapping $$\Phi:(0, \infty)\times(-\pi,\pi)\rightarrow \mathbb{R}^2\backslash (-\infty,0]\times\{0\}, \Phi (r, \theta) = (r ...
2 votes
1 answer
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How to find the derivative w.r.t. lower limits?

How to find $\frac{d}{dy} \int_{y}^{\infty} \int_{2y}^{\infty} y f(x_2)dx_2 f(x_1)dx_1$, where $x_1$ and $x_2$ are two independent continuous random variables and $f(x_1)$ and $f(x_2)$ are their PDFs. ...
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1 vote
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Limit of a Sequence of Definite Integrals

This was on a mock test for an examination that grants admission to an undergraduate course in mathematics. So, in theory, a school-going $17$ year old with a bit of extra knowledge should be able to ...
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3 answers
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Example for continuous $f$ such that $\int \limits_{0}^\infty |f(x)|dx < \infty$ and $\int \limits_{0}^\infty f^2(x)dx=\infty$. [closed]

Does $f = \dfrac{\sin(x)}{\sqrt{x}}$ make it? I know that $\int \limits_{0}^\infty \dfrac{\sin^2(x)}{x}dx = \infty$, but does $\int \limits_{0}^\infty \dfrac{|\sin(x)|}{|x|}dx < \infty$? What would ...
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1 answer
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Solve the integral: $\int_{0}^{\infty}\exp(-ax)dx$

I am trying to solve the integral: $\int_{0}^{\infty}\exp(-ax)dx$ Computing the indefinte integral gives: $-\frac{-1}{a}\exp(-ax)$ Now I was wondering if the following is correct: for $\infty$: $-\...
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Double Integral in Polar Form. Express $D$ in polar coordinates.

Let $D$ be the region bounded by $y=x^2$ and $y=2$. Express $D$ in polar coordinates. I would like to ask how to covert it to polar coordinates? How to make use of the two equations to know what is ...
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1 vote
2 answers
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Evaluate: $\int_0^2\int_0^{2-x}(x+y)^2 e^{2y\over x+y}dy dx$.

Evaluate: $$ \int_0^2\int_0^{2-x}(x+y)^2 e^{2y\over x+y}dy dx. $$ My attempt: I've changed the order and got this as $$ \int_{y=0}^2\int_{x=0}^{2-y}(x+y)^2 e^{2y\over x+y}dy dx. $$ and then ...
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Evaluating $\int_{0}^{\infty}x^{a}e^{-x^{3}}dx$ Using Spherical Coordinates

(Motivation) I made this improper integral for fun: $$\int_{0}^{\infty}x^{a}e^{-x^{3}}dx = \frac{1}{3}\Gamma\left(\frac{a+1}{3}\right)$$ where $\Re(a) > -1$. After struggling with various ...
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0 answers
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I Understand how to use chain rule and the power rule and how to use the chain within the power rule. NOT sure of which rule to apply sometimes. [closed]

I seem to misunderstand when to use/apply the chain vs Power rule. I know chain is for composite functions but I need more than that textbook definition. I need some more detailed explanations and ...
0 votes
1 answer
51 views

Evaluating the indefinite integral: $\int\sin^2(\sin(x))\cot(x)dx$

WolframAlpha gives me the solution that the integral $$\int \sin^2(\sin(x)) \cot(x) \, dx = \frac12 \log(\sin(x)) - \frac12\text{Ci}(2\sin x)$$ plus a constant, where $$\text{Ci}(y) = - \int_{y}^{\...
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1 answer
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How to do this financial math (Exam FM) question from first principles (summation)

A perpetuity costs 77.1 and makes end-of-year payments. The perpetuity pays 1 at the end of year 2, 2 at the end of year 3, ...., n at the end of year (n+1). After year (n+1), the payments remain ...
1 vote
0 answers
17 views

Double Integral Of A Sine Function Using Change of Variable

I have to calculate the integral $$\int_0^x \int_x^{2\pi} \left( \frac{ \sin(N(u-v)/2)}{\sin((u-v)/2)} \right)^2 du dv$$ in terms of $x$ and was given a hint to use change of variable substitution: $a ...
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2 votes
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How does this Dirac delta constrain the region of integration?

I have a function of two two-component vectors with this shape $$F(\vec{p},\vec{q})=G(\vec{p},\vec{q})\times\delta\Big(a+b(|\vec{p}|+|\vec{q}|)-\vec{v}\cdot(\vec{p}+\vec{q})\Big),\tag{1}$$ with $a>...
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integration question related to trigonometry [closed]

click me to see the question how can we solve this question? kindly suggest multiple ways if any....
2 votes
0 answers
36 views

Surface integral where $S$ is the surface of a solid bounded by cylinder $x^2+z^2=4$ and planes $y=0$ and $y=3$.

$$\iint_{S}\langle-3z^2,1-x,(y-2z)\rangle\,d\vec{s}$$ I am trying to solve this surface integral where $S$ is the surface of a solid bounded by cylinder $x^2+z^2=4$ and planes $y=0$ and $y=3$, I ...
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2 votes
1 answer
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Why can I rewrite this term as a root, but not the other?

I was practicing u-substitution. With the first problem, I was able to rewrite $u^{1/3}$ as the cube root of $u$, but when I did the same approach again with $u^{3/2}$ as the square root of $u^3$, ...
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1 answer
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Phantom Function $\xi(x)$ with antideritative equal $ 1$

There where some posts about antideritative of 0 but we know that there are infinitely many of them and they are equal to constant $+C$, which you have to not forget too add after indefinite ...
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5 votes
3 answers
134 views

Calculate the following integral $ \int\frac{2x+1}{x^{n+2}(x+1)^{n+2}}\ln\left(\frac{2x^2+2x+1}{x^2(x+1)^2}+\frac7{16}\right)dx$

Hello I am trying to solve a pretty complicated integral. It is a from a set of problems, published in a monthly journal for high school students and they are exercises in preparation for a ...
0 votes
1 answer
38 views

Kronecker delta in integral

I am interested in calculating the follwing integral $$I=\lim_{T\to\infty}\frac{1}{T}\int_0^Tdt\iint_0^\infty dE\;dE'e^{i(E-E')t}f(E,E'),$$ for a complicated function $f$. One might initially ...
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-2 votes
0 answers
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Finding center of mass of a sphere and cylinder by triple integrals and cartesian coordinates [closed]

I am required to prove the center of masses of a sphere and cylinder of constant densities by triple integration with cartesian coordinates. I know that using cylindrical coordinates makes the process ...
1 vote
0 answers
82 views

Layer Cake Representation Intuition

Let $(X, \mathcal{F}, \mu)$ be a measure space. If $f: X \to [0, +\infty)$ is non-negative and measurable, then $$ \int_X f(x) d\mu(x) =\int_0^\infty \mu(\{ x \in X: f(x)\geq t\})dt $$ It is not very ...
-1 votes
1 answer
66 views

How to find area bounded between $x=y^3-y$ and $x=0$ [closed]

Can someone help me with the integration so I can find the area between $x=y^3-y$ and $x=0$?
-2 votes
0 answers
33 views

Help with Conditional Probability Integral

All, I am trying to obtain the simplest expression for the following probability expression. Ultimately, I need the expression that results from the integral, but if there is a similarly nice ...
-2 votes
0 answers
37 views

Function for a beauty blender

Hi so I am trying to find a function to model my beauty blender (Ref: beauty blender). The closest I've gotten is the following function is \begin{align*} \sqrt{b^2-b^2kx-\frac{x^2b^2}{a^2}+\frac{x^3b^...
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3 votes
3 answers
111 views

Integral power of error function with exponential

I'm trying to solve the integrals below: $$ \int_{-\infty}^{\infty}dx \hspace{5pt}\mathrm{erf}\left(x b\right)^2\exp\left(-x^2a\right) ,$$ with $a,b>0$. Unfortunately, I could not find this ...
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0 answers
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Show that lim n → ∞ ∫ n 0 ( n + x n ) n e − 2 x d x = 1 [closed]

Can anybody help me with this question
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0 answers
29 views

How can i solve this disequality? Integral and expected value

I would like to demonstrate that when Y>X, this inequality is true. $E_{y}+\left(\int_{a}^{+\infty}\left(y-E_{y}\right)^{m}f_y(y)dy\right)^{\frac{1}{m}}\geq E_{x}+\left(\int_{b}^{+\infty}\left(x-E_{...
3 votes
1 answer
26 views

Deriving marginal distribution from a joint distribution

Let the joint distribution of a random vector be $$f(x,y)=\begin{cases}1,\enspace 0<x<1, x<y<x+1&\\0, \enspace {\rm otherwise}\end{cases}$$ The marginal pdf of $X$ is $Uni(0,1)$, but ...
9 votes
1 answer
207 views

Can we evaluate $\int \frac{1}{\sin ^ {2n+1} x+\cos ^ {2n+1} x} d x?$

After investigating the integral $$\boxed{\quad \int \frac{1}{\sin ^5 x+\cos ^5x} d x \\=\frac{4}{5}\left[-\frac{1}{\sqrt{2}} \tanh ^{-1}\left(\frac{\sin x-\cos x}{\sqrt{2}}\right)+\frac{1}{\sqrt{\...
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0 votes
0 answers
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Analytic continuation of $\int_{-\infty + i \cdot p}^{\infty + i \cdot p} \exp\left[ -w^{2} +\left|w-y- i\cdot p\right|-y^{2}\right]\operatorname{d}w$

I am dealing with the following integral defined in Eq. $(40)$. I am kind of sure that steps from Eqs. $(40)$ to $(41)$, $(41)$ to $(42)$ are both fine. I tested on Mathematica step from Eq. $(42)$ to ...
  • 41
1 vote
1 answer
108 views

$\alpha(b) - \alpha(a) \geq \int_a^b \alpha'(x) dx$ for $\alpha$ monotone and differentiable a.e.

This is a problem from Reed and Simon's book on functional analysis, where you are asked to prove $\alpha(b) - \alpha(a) \geq \int_a^b \alpha'(x) dx$ for a monotone function $\alpha$ that is ...
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0 votes
3 answers
50 views

Problem with integration by substitution - can't understand how $u$ was chosen.

This is the problem and the solution to it: $$\begin{split} \int \frac{1}{x-\sqrt{x}}dx& \\ u=\sqrt{x}-1&\quad du=\frac{1}{2\sqrt{x}}dx\\ \int \frac{1}{x-\sqrt{x}}dx &=2\int\frac1u du\\ &...
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1 answer
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How to show that the integral of$ \frac{1}{x-x\log x}\mathrm{d}x $ from $ 0$ to $1$ diverges?

I have a problem regarding $$\int_{0^+}^1 \frac{1}{x-x\log x}\mathrm{d}x.$$ I think the integral does not converge. How can I show this?
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+50

Numerical integration: The composite Newton-Cotes formulas, uniqueness and inductive definition for a given order of exactneness

I have a question on Rabinowitz and Davis: Methods of numerical Integration. They start to give a sequence for what they call The (composite) Integration Newton-Cotes formulas. This together with my ...
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-1 votes
0 answers
38 views

Is the integral convergent? [closed]

I tried to find if this integral converges and got to the conclusion that it does. However I cannot be sure because I dont have answers for that problem (took it from an exam) and I'll be happy if ...
0 votes
1 answer
39 views

Derivation of the self-adjoint method for solving neural ODE

Let $z\in R^N$ be the vector in the ODE: $\frac{d z}{d t}=f(z,\theta,t)$. $\theta$ is the parameter for the neural network and $L:R^n\rightarrow R$ is the loss function. More details for the problem ...
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3 votes
1 answer
133 views

How to solve $\int \frac{f(x)}{f^{\prime}(x)} dx$

Let $f:\mathbb{R}\to\mathbb{R}$ be of class $C^{\infty}$ with $f^{\prime}\not\equiv0$. There exists a formula to solve the integral $$\int \frac{f(x)}{f^{\prime}(x)} dx?$$ Since I know that is $$\int \...
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1 vote
0 answers
31 views

How to compute this integral (over a Lie group)?

I met the following question and I don't know how to compute it directly: Let $G:=GL(2,\mathbb{R})$, and equip $G$ with the Haar measure $$dg:=\frac{1}{|det(g)|^2}dg_{11}...dg_{22},~(g:=(g_{ij})).$$ ...
-4 votes
0 answers
22 views

I know a function with finite number of discontinuity points is integrable, but this question is different [closed]

Let f be a bounded function on a bounded interval [a, b], and f is continuous on [a, b] except at a sequence of points {an : n = 1,2,···} ⊂ [a,b]. Assume limn→∞ an = c. Show that f is integrable over [...
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1 vote
1 answer
99 views

Integral of $\int f^2(x)\,dx$

Can we write the integral $$\int f^2(x)\,dx$$ in terms of $f(x)$, $x$ and $\int f(x)\,dx$? I've tried solving the integral by parts, but I can only end up with an equation of the form $0=0$. Maybe it ...

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