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Questions tagged [leibniz-integral-rule]

Also known as Feynman's trick or differentiation under the integral sign.

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Derivative of double(multiple) integrals

How do I find the derivative of function f with respect to t: $$ f(t) = \int_{ }^{ }\int_{ }^{ }\sqrt{x^{2}+y^{2}}dxdy $$ and the domain of integral is: $$ (x-t)^2 + (y-t)^2 \leq 1 $$
Ali Daemi's user avatar
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Derivative of Lebesgue integral with indicator functions

Suppose we have a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$. I want to take the derivative of $$ W\left(x\right)\equiv\int_{\Omega}\mathbf{1}\left\{ \omega\in R\left(x\right)\...
Thomas's user avatar
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Can you solve $\int_0^1\!\sqrt{x^4+4x^2+3}\,\mathrm{d}x$ using Feynman's differentation under the integral sign trick? [closed]

My question is what stands in the title. Can you solve $\displaystyle\int_0^1\sqrt{x^4+4x^2+3}\,\mathrm{d}x$ using Feynman's differentation under the integral sign trick?
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Evaluate the derivative of the integral to get the specific form

$$ \frac{d}{dt}{\alpha}\left(t\right)=\frac{d}{dt}\left[2 \int_{t}^{\infty} \frac{d x}{x \sqrt{\left(\frac{x}{t}\right)^2\left(1-\frac{1}{t}\right)-\left(1-\frac{1}{x}\right)}}-\pi\right] $$ $$ ...
Aniruddha Ray's user avatar
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Derivative of $ f(x)=\int_{1 / x}^{e^x / x} \frac{\cos (x t)}{t} d t $

Find the derivative of the function $$ f(x)=\int_{1 / x}^{e^x / x} \frac{\cos (x t)}{t} d t, \quad(x>0) . $$ I use $\frac{d}{dx}\int_{1 / x}^{e^x / x} \frac{\cos (x t)}{t} d t=\frac{x\cos(e^x)}{e^x}...
bajsmackan's user avatar
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Trying to solve $I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx$

While trying to find an answer to this problem on the forum, I came across this integral: $$ I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx \tag 1$$ Where $c$ and $k$ are real numbers. I ...
FriendlyNeighborhoodEngineer's user avatar
1 vote
1 answer
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Leibniz integral rule help

Let $$I:=\frac{\partial}{\partial \epsilon} \left[\epsilon \int^{b(\epsilon)}_{a(\epsilon)} x f(x) dx \right]_{\epsilon = 0}.$$ My textbook claims that, as a consequence of the Leibniz integral rule: $...
lohey's user avatar
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Reasonable and Logical Integration

$\lim _{k\to +\infty}$ $\int_{0}^{k[x]} (kt-[kt])^k dt$ $;$ $k\in N$ is $[\frac{\lambda x} 2]$ Where $[.]$ denotes greatest Integer function then the value of $\lambda$ would be? My approach to this ...
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A peculiar integral using Leibnitz rule of integration?

Mathworld while explaining (rather very briefly) the Leibnitz rule of integration, aka derivative under an integral sign, mentions that this method may be used to evaluate peculiar integrals such as $$...
Firdous Mala's user avatar
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Variable limits (function of x) in a definite integral

I am stuck in a concept of Definite Integration. Let us say we have a function that goes like ${f(x) = \int_0^x e^{x-t} f(t) \,dt}$ Now I wanted to know that if I put $x=0$, will the limit range from $...
Aayush's user avatar
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2 answers
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partial derivatives and integrals

suppose I have a function $$v(x,t):=\int_0^t u(x,t;s) ds.$$ Why $$v_t(x,t)=u(x,t;t)+\int_0^t u(x,t;s) ds.$$ I do not know where does the $u(x,t;t)$ term come from ?
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6 votes
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Evaluating $\int_1^{\sqrt2} \frac{\tanh^{-1}(\sqrt{2-x^2})}{1+x} dx$

I was trying to compute the value of the integral $$\ I = \int_1^{\sqrt2} \frac{\operatorname{arctanh}(\sqrt{2-x^2})}{1+x}dx$$ I began by declaring the family of integrals: $$\ I(a) = \int_1^{\sqrt2} \...
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Two-dimensional generalization of Leibniz's integral rule

Given a function $f(x,y):\mathbb R\times \mathbb R\to \mathbb C$ and a real parameter $\theta$, one can use Leibniz's integral rule to solve \begin{equation}\label{eq}\tag{1}\frac{d}{d\theta}\int_{a(\...
A Quantum Field Day's user avatar
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1 answer
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How to Calculate the Partial Derivative F_u(1,v) of the Integral Function F(u,v)

I'm working on a problem in calculus and am having difficulty with a specific function and its partial derivative. The function is defined as: $F(u,v) = \int_{uv}^{u+v}e^{-(u-y)^2}dy$ I'm trying to ...
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Help with Variational Calculus & Leibnitz Rule

Let $f: (0,\infty)\to \mathbb{R}$ be convex and lower-semicontinuous with $f(1)=0$ and $\mu$, $\hat{\mu}$ be two probability distributions on a measurable space $\mathcal{X}$ which are absolutely ...
pablopez's user avatar
5 votes
1 answer
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How to solve $\delta''(\vec{k},\tau)+\mathcal{H}(\tau)\delta'(\vec{k},\tau)-\dfrac{3}{2}\Omega_m(\tau)\mathcal{H}^2(\tau)\delta(\vec{k},\tau)=0$

While self-studying a set of notes about Cosmology, I have encountered the following claim: We have the linear growth equation: $$\delta''(\vec{k},\tau)+\mathcal{H}(\tau)\delta'(\vec{k},\tau)-\dfrac{...
Wild Feather's user avatar
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I am having trouble with applying Leibniz Rule for Differentiation an integral.

I have a function, $V(D) = (1-\tau_E) (1-\tau_C) \int_0^{100} (R-D) dF(R) + (1-\tau_D) \int_D^{100} D dF(R) + (1-\tau_D)(1-b) \int_0^D RdF(R)$ Where R is a random variable $R \in [0,100]$ with ...
closed_form's user avatar
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Differentiation under integral sign - problem while finding the constant c

Evaluate: $$\phi(y) = \int_0^\frac{\pi}{2} \frac{1+y\sin^2(x)}{\sin^2(x)} dx $$ After applying the Leibniz formula, the indefinite integral of $d\phi(y)$ comes out to be: $$\int d\phi(y)=\frac{\pi}{2}\...
Spencer's user avatar
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How to apply Leibniz rule for the derivation of an integral when the integration variable depends on the parameter

I know that, according to the statement of the Leibniz rule for the derivation of a parametric integral, if we have: $$F(\lambda)=\int^{b(\lambda)}_{a(\lambda)}f(\lambda,x)dx$$ then the following ...
Wild Feather's user avatar
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A scalar is equal to a vector? Leibniz rule of integration for multivariate Gaussian

The density of the multivariate Gaussian (=normal) distribution is given by $$ f(\mathbf{x},\boldsymbol{\mu},\Sigma)=\frac{1}{(2\pi)^{\frac{d}{2}} \left|\Sigma\right|^{\frac{1}{2}}}e^{-\frac{1}{2}(\...
mto_19's user avatar
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On the PDF of a function of a random variable [duplicate]

Suppose I have a random variable $X$ and I know its PDF $p_X$. The goal is to find the corresponding $p_Z$ of the random variable $Z=f(X)$. For now, $f$ is any Borel measurable function. I'm trying to ...
haiku's user avatar
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2 votes
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Evaluate: $ F(\alpha) = \int_0^1 \frac{x^\alpha-1}{\ln x} dx $. Different answers from different approaches.

Why is approach 2 wrong here? I am relatively newe to calculus please explain in detail what is going wrong... Evaluate: $ F(\alpha) = \int_0^1 \frac{x^\alpha-1}{\ln x} dx $ Approach 1: (Correct) ...
Akshit Chhabra's user avatar
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1 answer
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question on Leibniz formula

let $$f(x)=\int_{1}^{x}{\frac{\cos y\sin x}{y^2+y+1}dy}$$ then find all values of $x$ for which $f'(x)=0$? I tried directly differentiating but I got stuck on $$f'(x)=\cos x\left(\int_1^x{\frac{\cos y}...
Mandar Ajinthekar's user avatar
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Prove that a derivative of an integral is constant using the Leibniz rule

I analyse the proof that the derivative of the function below exists and is zero, and that the function is constant itself. Let's define $$f(u) = \int_{-\infty}^\infty e^{{u^2\over2}-{x^2\over2}+iux} ...
c00mmenter's user avatar
4 votes
1 answer
260 views

Leibniz integral rule for an arbitrary number of dimensions

Suppose we have $N$ particles whose coordinates are given by $\mathbf{r}_{i}$. These coordinates are confined to be within a three-dimensional unit cell defined by $$\mathcal{V}=\left[0,L_{x}\right]\...
user186483's user avatar
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Time derivative of integrals on changing level and upper contour sets

There are related questions and answers on StackExchange but I cannot find a general formula for the following types of (time) derivatives of integrals with respect to changing level and upper contour ...
Confusian's user avatar
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1 answer
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Intuition for Leibniz's integral rule

Let $g(x)$ be the density of some random variable $x$. I am looking at the following expected value: $\int_a^b f(x, a) g(x) dx$. I am curious what happens if I upwards shift the lower bound: $$ \frac{...
FooBar's user avatar
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Is a complex continous function of $f$ on a disk around the origen also continous as a function of $\bar{z}$?

I am trying to use the Leibnitz theorem on a continous funktion $f: B \rightarrow \mathds{C}$, with $B$ being an open disk around the origen of Radius $R$, on $f$ as a function $f(z, \bar{z})$. It is ...
PanicStation's user avatar
1 vote
2 answers
132 views

About the integral $\int_0^{\pi/2} (\sin x-1)/\ln(\sin x) \mathrm{d}x$

Using the Feynman Technique, Let $$I(a) := \int_0^{\pi/2} \frac{(\sin x)^a - 1}{\ln(\sin x)} \mathrm{d}x$$ $$I’(a)= \int_0^{\pi/2} (\sin x)^a \mathrm{d}x = \frac{\sqrt{π}}{2} \frac{\Gamma\left(\...
integral's user avatar
2 votes
2 answers
97 views

The integral $I(a,b)=\int_0^1x^a\log^b(x) \space dx$

I found a formula for the integral $$I(a,b)=\int_0^1x^a\log^b(x) \space dx$$ By just differentiating the integral $I(t)=\int_0^1x^t\space dx $ under the integral sign b times and evaluating the ...
Bilge K. Aksebzeci's user avatar
1 vote
0 answers
86 views

How is Leibniz integration rule applied to stochastic integrals? [duplicate]

I am working through the mathematics behind the Hull-White short rate model and am currently stuck on how to take the partial derivative of and evaluate a stochastic integral when looking at how bond ...
Learner248079's user avatar
2 votes
1 answer
101 views

Differentiation of an improper parametric integral

In a textbook on fractional calculus, the following equality is stated without any comments or a proof: $$ \frac{d}{dt} \left( \int\limits_{-\infty}^{t} (t-s)^{-\alpha} f(s) ds \right) = \alpha \int\...
AnonymousUser's user avatar
1 vote
2 answers
366 views

Leibniz's rule application for exponentials

Let $\displaystyle f(x) = \int_{0}^{x} e^{\frac{x^2}{2}-\frac{t^2}{2}} dt$. Show that $f'(x) - x \cdot f(x) = 1, ∀ x \in \mathbb{R}$. I'm pretty sure I'm supposed to use Leibniz's rule for this one, ...
Arthur De Arola Brito's user avatar
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1 answer
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Can I change the order of integration when the upper limits are infinite?

I'm trying to solve this : $$\frac{d}{dy}\int_{y}^{\infty} \int_{f(y)}^{\infty} (g(y)+h(x_2))f(x_2)dx_2 f(x_1)dx_1$$ In this case, can I change the order of Integration ? I will get this : $$\frac{d}{...
ycole's user avatar
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Applying Leibniz rule for double integral

I'm trying to solve A = $\frac{d}{dy} \int_{0}^{y} \int_{0}^{k(x_1,y)} (g(x_1)+h(x_2))f(x_2)dx_2 f(x_1)dx_1$. Here is my approach: let $j(x_1, y) = \int_{0}^{k(x_1,y)} (g(x_1)+h(x_2))f(x_2)dx_2$, then ...
ycole's user avatar
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1 answer
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Use Leibnitz Theorem to solve the following- [closed]

If anyone could break it down to smaller steps, I'd appreciate alot.
shreyash1611peep's user avatar
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Leibniz integral rule for the Airy function.

We recall that the Airy function is (well) defined by: $\text{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + xt\right) dt$ I want to apply the theorem of differentiation under the ...
Francis Benjamin's user avatar
2 votes
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The Leibniz integral rule

I've had problems applying the Leibniz Rule. The professor's slides say: Consider the following value function \begin{equation} \nu(t) = \int_{t}^{\infty} e^{- \int_{t}^{s}r(z)dz} \pi(s) ds \end{...
Maximilian's user avatar
1 vote
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A proof of Leibniz integral rule

I'm reading Leibniz integral rule Let $X$ be an open subset of $\mathbb R^d$ and $(\Omega, \mathcal F, \mu)$ a measure space. Let $f:X \times \Omega \to \mathbb R, (x, \omega) \mapsto f(x, \omega)$. ...
Analyst's user avatar
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4 votes
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Is the Local Average of a Continuous Multivariable Function Differentiable?

Suppose we have a continuous $f:\mathbb{R}\to\mathbb{R}$. It is an immediate corollary of the Leibniz integral rule that $f^*:x\mapsto\int_{x-\frac{1}{2}}^{x+\frac{1}{2}}f(t)\ dt$, the "local&...
Thomas Anton's user avatar
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2 answers
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Leibniz Integral Rule applied n-times

I was trying to come up with a generalization of the Leibniz Integral Rule $$ \frac{d}{dx}\left(\int_a^xf(x,t) \; dt\right) = f(x,x) + \int_a^x \frac{\partial}{\partial x} f(x,t) \; dt $$ and ...
elson1608's user avatar
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Reynolds transport theorem: link between different formulations

Currently I am studying the reynolds transport theorem. From my research I could find two depictions of the theorem. My problem lies withing tying these two formulations together. "Mathematical/...
Manuel's user avatar
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2 votes
2 answers
136 views

Fundamental theorem of calculus with the derivative on the inside?

I know that: $$\frac{d}{dx}\int_{a}^{g(x)} f(t)dt = f(g(x))*g'(x)$$ But what about: $$\int_{a}^{g(x)}\frac{d}{dx}f(x)dx$$ An example of this would be: $$\int_{3}^{t^3}\frac{d}{dx}\frac{x}{x-2}dx$$ Do ...
Omar Aboutaleb's user avatar
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Leibniz formula for integral with spherical coordinates in multi-dimensional space

Given $u: \mathbb{R}^n \to \mathbb{R}$ smooth enough and set $\bar{u}(r) = \int_{|x| = r}u(x)d\sigma$, with $x = \left(r, \sigma\right)$ is the spherical coordinate of $x$. I have some differences in ...
Larry Baynes's user avatar
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1 answer
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the Bayes rule with a general form (Young textbook problem 3.3)

Hi so I am trying to find the bayes rule with a loss function defined as for a general estimator. we just need to find the general form of the Bayes rule for this loss function. let the prior be ...
sophie-germain's user avatar
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1 answer
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Can we prove the following by using Leibniz's rule for differentiation of a definite integral?

Let $\Phi(t)$ be $$\Phi(t)=\int\limits_t^\infty \kappa e^{-\int\limits_t^s g(\omega)d\omega} ds$$ such that $0<t<s$. If it holds that $\Phi(t)=\alpha$ where $\alpha$ is constant $\forall t$ can ...
Oliver Queen's user avatar
2 votes
1 answer
52 views

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. ...
ycole's user avatar
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0 votes
1 answer
128 views

Derivative of the integral of $\exp(ax)\over 1+\exp(x)$

Given $a\in(0,1)$ I want to prove the function \begin{align*} g(a)=\int_{\mathbb{R}}\frac{\exp(ax)}{1+\exp(x)}dx \end{align*} is infinitely differentiable. I proved the function $f(a,x)=\frac{\exp(ax)}...
RyeCatcher's user avatar
1 vote
2 answers
77 views

Evaluate a derivative of the function $f(x,y)=(\int_0^{x+y} \varphi, \int_{0}^{xy} \varphi)$

The problem is the following. Evaluate the derivative of the function $f(x,y)=(\int_0^{x+y} \varphi, \int_{0}^{xy} \varphi)$ in the point $(a,b)$, where $\varphi$ is integrable and continuous. The ...
Brien Navarro's user avatar
-2 votes
2 answers
82 views

How to integrate using differentiation under the integral sign

$$\int _0^{\infty }x^2e^{-ax^2}dx$$ where $a > 0,$ given that $\displaystyle\int _0^{\infty }e^{-ax^2}dx=\frac{1}{2}\sqrt{\frac{\pi }{a}} $. Not sure where to begin with this
jalkdsji's user avatar