Questions tagged [leibniz-integral-rule]

Also known as differentiation under the integral sign.

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Derivatives of integrals with Variable Bounds

I'm working through a Green's function problem for a second-order linear differential operator $L$ and I need to find the derivative of: $$ u(x) = \frac{u_2(x)}{a_0(\xi)W(\xi)} \int_a^x u_1(\xi)f(\xi) ...
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Leibnitz integral rule on double integral

I am trying to compute the following partial derivative: $$ \frac{\partial}{\partial x}\iint_{\Omega(x)} f(x,\mathbf{w})d\mathbf{w} $$ with $f:\mathbb{R}\times \mathbb{R}^2\to \mathbb{R}$. In this ...
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How to find $F^{(n+1)}(x)$ if $F(x)=\int_0^x(x-t)^nu(t)dt$?

if $F(x)=\int_0^x(x-t)^nu(t)dt$ then find $F^{(n+1)}(x)$ from Leibniz rule $a(x)=0,b(x)=x,a'(x)=0,b'(x)=1$ and $G'(x)=n(x-t)^{n-1}\\$ so $F^{(1)}(x)=n\int_0^x(x-t)^{n-1}u(t)dt$ and $F^{(n)}(x)=n.(n-1)....
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Let $F\left(x\right)=\int_{t^2}^{x^3}{\frac{\mathrm{d}y}{\sqrt{x^2+y^4}}}$. Find $F'\left(x\right)$

Let $$F\left(x\right)=\int_{t^2}^{x^3}{\frac{\mathrm{d}y}{\sqrt{x^2+y^4}}}$$ Find $F'\left(x\right)$ Had the lower limit been a function of $x$ I would have easily calculated. I am not very familiar ...
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Leibniz rule for high dimension

I need to differentiate an integral using Leibniz rule, but I encountered a problem. I will state it in the simple case. Suppose I want to compute the derivative of $ \int \|x\|^2 t dt$ with respect ...
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1answer
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Leibniz rule X Direct integral computation

This question arose on a very complicated context, but it is very simple: I have for $0<a<1$ the integral $\int_0^t (t-x)^{a-1} dx$ and 'd like to derive on $t$. The imediate result is $t^{a-1} $...
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Two variable definite integral

Q) Let $f:R^+\rightarrow R$ be differentiable function with $f(1)=3$ and satisfying $$\int_{1}^{xy}f(t)dt=y\int_{1}^{x}f(t)dt+x\int_{1}^{y}f(t)dt; \forall x,y\in R^+$$, then $f(e)=?$ My Attempt:I ...
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Do you know any result on how to interchange stochastic integration with Frechet derivative?

Let we have a stochastic basis $(\Omega,\mathcal{F},\mathbb{F} = (\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ with standard assumptions. Let the integral $\int_0^t f(s,h) dB_s$, for $f(s,h)$ a ...
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How to show two identical integrals $\; 2\int_{-r}^{r}f(x)dx=r\int_{-r}^{r}\sqrt{1+(df/dx)^{2}}dx$

With $$ \;f(x)=(r^2-x^2)^{1/2}$$ show that$$\; 2\int_{-r}^{r}f(x)dx=r\int_{-r}^{r}\sqrt{1+(df/dx)^{2}}dx$$ by using integration by parts. Then, conclude that the area of a circle of radius r is $\;\pi ...
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Solving $\int_0^{\pi/2}x\cot x\,\mathrm{d}x$ while running into a $0\times\infty=0$ problem

While i have been trying to solve the integral $ \int_0^{\pi/2} x\cot x \, \mathrm{d}x $ i have noticed that by trying integrating by parts using $u = x$ and $\mathrm{d}v = \cot x \, \mathrm{d}x$, i ...
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Proving the Leibniz Integral Rule

I've been wondering if there is a fairly simple proof or derivation for the following (called the Leibniz Rule): $$ \frac{d}{dt} \int_{a(t)}^{b(t)}f(x,t)dx = \int_{a(t)}^{b(t)}\frac{\partial{f}}{\...
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Show that $\int_0^\pi \log( 1 - 2r\cos(t) + r^2)\, dt=0$

Show that for $r \in (-1,1)$ $$ \int_0^\pi \log( 1 - 2r\cos(t) + r^2)\, dt = 0$$ Here's what I did so far: $$f(r,t) = \log(1 - 2r\cos(t) + r^2) = \log( (1-re^{it})(1-re^{-it}))$$ The Leibniz rule ...
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Computing An Integral Using Leibnitz Rule

I am studying dynamical systems on my own and working through a brief introduction of Poincare Maps. In my text the author takes the derivative of $(*)$ with respect to $x_0$, and I believe this is ...
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47 views

A step in deriving the solution to heat equation

I'm trying to understand how to solve the heat equation. I'm stuck on this particular step: $$Q(x,t) = \dfrac{1}{2} + \dfrac{1}{\sqrt\pi}\int_{0}^{\dfrac{x}{\sqrt{4kt}}}e^{-s^2}ds, t > 0.$$ I want ...
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Trouble with $I(\alpha) = \int_0^{\infty} \frac{\cos (\alpha x)}{x^2 + 1} dx$

I'm ultimately trying to solve $$I(\alpha) = \int_0^{\infty} \dfrac{\cos (\alpha x)}{x^2 + 1} dx$$ by using differentiation under the integral. I realize that this is most easily done using residues ...
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1answer
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Using Dominance Convergence Theorem in derivation of Leibniz's rule

I really need help understanding how Dominance Convergence Theorem applies to the following limit: $$ \underset{\Delta t\rightarrow 0}{lim}\int_{a(t)}^{b(t)}\frac{f(x, t+\Delta t)-f(x,t)}{\Delta t}dx $...
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$\int_{e}^{\infty} \frac{x+1}{x^3\ln ^2(x)} dx$

I am trying to solve the given integral (on my own, not a homework problem.) $$\int_{e}^{\infty} \frac{x+1}{x^3\ln ^2(x)} dx$$ I am having problem with a part of using Feynman's technique which I need ...
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Leibniz rule - problem with one of the parts of the function when it is not defined at some x

I'm having trouble with solving the following integral with Leibniz's rule. $F\left(\alpha\right) = \int_{0}^{π}{\frac{\ln{\left(1+\sin{\alpha}\cos{\alpha}\right)}}{\cos{x}}\,\mathrm{d}x} $ The ...
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Can't use Leibniz integral rule

So I've noticed a mistake in my probability text book It's clearly wrong in the integral part, which is how to deal with integral below $$\int_{0}^{2\pi}\frac{1}{1-\rho\sin(2\theta)}\,d\theta\qquad({-...
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Evaluating $\int_{0}^{1}\frac{(x^{\phi}-1)^2}{(\ln x)^2}\mathrm dx$

$$\int_{0}^{1}\frac{(x^{\phi }-1)^2}{(\ln x)^2}\mathrm dx$$ This question comes from here. I tried it using the same obvious method as the solution provided, but I do not seem to be getting the same ...
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1answer
94 views

Derivative of integral in one direction

Suppose I have a 3D scalar field $\phi(x,y,z)$, where $x,y,z$ are cartesian coordinates in $\mathbb{R}^3$. I can define another 3D field, which is the integral of the field with respect to one ...
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Leibniz rule, problems with matrix entries.

I have the next: Let be $U \subset \mathbb{R}^{m}$ open, $\phi: U\times [a,b] \rightarrow{\mathbb{R}^{n}}$ with partial derivate continuos $\partial_{1}: U\times [a,b] \rightarrow{T(\mathbb{R}^{m}, \...
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Can the Leibniz integral rule be applied to an integral whose integrand does not contain the variable of derivation?

To my understanding, the Leibniz Integral lets $$\frac{d}{dx}\int_{a(x)}^{b(x)} f(x,t)dt = \int_{a(x)}^{b(x)}\frac{d}{dx}f(x,t)dt$$ under certain restrictions. (Pretend the second $\frac{d}{dx}$ is a ...
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A problem related to $\int_{0}^{1} f(x)(x-f(x))dx=1/12$

Consider a differentiable function satisfying $$\int_{0}^{1} f(x)(x-f(x))dx=1/12$$ Then find the nearest integer less than or equal to $\frac{1}{f'(1)}$. Let $$F(x)=\int_{0}^{x}f(x)(x-f(x))$$ we ...
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1answer
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Improper integral $\int_{0}^{\infty}\tan^{-1}(\alpha x) \tan^{-1}(\beta x)/ x^2 dx, \alpha, \beta>0.$

I am facing difficulty in evaluation of the following improper integral: $$\int_{0}^{\infty} \frac{\tan^{-1}(\alpha x) \tan^{-1}(\beta x)}{ x^2} dx,\ \ \alpha, \beta>0$$ I thought of applying ...
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General Leibniz rule (differential) notation using multi-index notation

When I use multi-index notation $\alpha = (\alpha _1, \alpha _2, \cdots , \alpha _n), \beta = (\beta _1, \beta _2, \cdots , \beta _n)$, Leibniz rule (partial differential) is described as follows. \...
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About General Leibniz rule (differential)

https://en.wikipedia.org/wiki/General_Leibniz_rule This site describes General Leibniz rule. At the content-4 Multivariable calculus, \begin{equation} \partial ^\alpha (fg) =\sum_{\beta : \beta \leqq \...
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First order condition of Lagrange and Integral.

I'm study Economics and I have a math question about the first order condition on an integral. Let's me explain: First, the Lagrangian for the problem (Optimal Allocation of Consumption Expenditures ...
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1answer
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Derivative of integral involving differential of CDF

I am struggling to compute the following partial derivative of an integral $\dfrac{\partial }{\partial t} \int_{a}^{\infty} x(t) dF(x(t)) $, where x is a random variable that depends on the ...
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Newton Leibniz Theorem

These both formula came under Newton Leibniz Theorem. But i don't understand when to use the formula '1.' and when the formula in '2'. I was trying to solve this question. In this question we have to ...
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1answer
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Evaluate the limit $\lim_{x \to 0+}\int_{x}^{2x}\frac{\sin^mt}{t^n}dt$

Evaluate the limit $$\lim_{x \to 0+}\int_{x}^{2x}\frac{\sin^mt}{t^n}dt$$ where m ,n are natural numbers. Take cases when $m\ge n$,,when $n-m=1$,when $n-m>1$ I started out by multipying numerator ...
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Leibniz's rule for $J(x) = \int_{t_0}^T h(x(t),t)dt$

Consider the following functionals: $$J(x) = \int_{t_0}^T h(x(t),t)dt$$ where $x(t) \in \mathbb{R}^n$, and $J(\cdot)$, $h(\cdot)$ are both real scalar functionals. I read the textbook ...
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Application of Leibniz's Integral Rule

Consider the following image showing the temperature of a metal pipe: The temperature $T(z,t)$ is a function of the length coordinate $z$ and time $t$. Integrating the partial derivative of $T$ with ...
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1answer
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Differentiation of definite integral with respect to function inside integrand

I have an integral which is of the form: $$ I = \int_0^\infty g(t, x(t)) \, dt. $$ I'm trying to demonstrate how incremental changes in $x(t)$, for any given $t$, affect $I$. Informally/intuitively, I'...
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a queer problem on repeated root

if $f(x),g(x),h(x),\phi(x)$ are distinct polynomials and let $$l(x)=\left(\int_{1}^{x}f(x)h(x)dx\right)\left(\int_{1}^{x}g(x)\phi(x)dx\right)-\left(\int_{1}^{x}f(x)\phi(x)dx\right)\left(\int_{1}^{x}g(...
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finding a function given a complicated relation involving integrals

Find a function $g$ continuous in $[0,\infty]$ and positive in $(0,\infty)$ satisfying $g(0)=1$ and $$ \tag{1} \frac{1}{2}\int_{0}^{x} g^2(t)dt=\frac{1}{x}f^2(x), $$ where $f(x)=\int_0^x g(t)dt$. My ...
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Let $f: \mathbb{R} \to \mathbb{R}$, continuous and bounded function

$\blacksquare$ Problem: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous and bounded function such that $$ x \int_{x}^{x + 1} f(t) \mathrm{d}t = \int_{0}^x f(t) \mathrm{d}t \quad \text{for any } x \...
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how to solve $\int _0^1\frac{\ln \left(1+x\right)}{a^2+x^2}\:\mathrm{d}x$

how to solve $$\mathcal{J(a)}=\int _0^1\frac{\ln \left(1+x\right)}{a^2+x^2}\:\mathrm{d}x$$ i used the differentiation under the integral and got \begin{align} \mathcal{J(b)}&=\int _0^1\frac{\ln \...
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Leibniz Integral Rule for Square Root of a function

Suppose that $ G(x,\alpha,\beta) $ has two simple roots at $ r_1, r_2 \in \mathbb{R} $, with $ 0 < r_1 < r_2 $. It is continuous on $ [r_1,r_2]$. We are trying to compute the derivatives of $$ \...
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1answer
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A question on application of derivatives

If the area of the circle increases at a uniform rate, show that the rate of increase of the circumference varies inversely as the radius. My approach- If area is increasing at a uniform rate then ...
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Can I replace the $\frac{d}{dt}$ with $\frac{\partial}{\partial t}$ in the Leibniz integral rule?

Leibniz integral rule can be applied like following if $N$ is a function of $t$ $$ \frac{dN}{dt} =\frac{d}{dt}\left(\int_{a(t)}^{b(t)}\rho(x,t)dx\right) $$ $$ \frac{dN}{dt} =\int_a^b \frac{\partial \...
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61 views

Solve the following DE and hence find $f(x)$

Solve: $$ f(x)=\left(1+x^{2}\right)\left[1+\int_{0}^{x} \frac{f^{2}(t)}{1+t^{2}} d t\right] $$ I have tried to solve this question by diffrentiating both sides with the help of $Lebnitz$ integral rule ...
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Correct application of Leibniz Rule

Consider the following integral $$I = \int^b_af[x[t];c]dt$$ where $a,b$ and $c$ are scalers and $x[t]$ is a function of $t$. I want to know $\frac{\partial I}{\partial x}$. If I apply Leibniz Rule, I ...
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Prove that $c_{m} \in[a, b],$ for all $m \geq 1, \lim _{m \rightarrow \infty} c_{m}$ exists and find its value.

Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function. Let $$ a = \inf_{0 \leq x \leq 1} f(x) ~\text{ and }~ b = \sup_{0 \leq x \leq 1} f(x) . $$ For every positive integer $m$, define $$ c_{m}...
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1answer
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Calculus - Leibniz Integral rule problem check

From Thomas Calculus we are given the following integral to differentiate Applying Leibniz Integral rule I got the following result $-2x\sqrt{x^6+x^2} - \int_1^{x^2}\frac{x}{\sqrt{t^3+x^2}}dt$ ...
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114 views

Find the value of $\lim _{a \to \infty} \frac{1}{a} \int_{0}^{\infty} \frac{x^{2}+a x+1}{1+x^{4}} \cdot \tan ^{-1}\left(\frac{1}{x}\right) \,d x $

Find the value of : $$ \lim _{a \rightarrow \infty} \frac{1}{a} \int_{0}^{\infty} \frac{x^{2}+a x+1}{1+x^{4}} \cdot \tan ^{-1}\left(\frac{1}{x}\right) \,d x $$ I have tried to evaluate this integral ...
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Swapping integral and derivative in >2 dimensions

Consider a function $f: \Omega \times \mathbb R^n \to \mathbb R$, where $\Omega$ is a measure space. Suppose for any $x \in \mathbb R^n$, $f(\omega, x)$ is differentiable for almost all $x$. I would ...
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60 views

Solve the following differential equation:

Find $g(x)$ from the following condition: $${g(x)}=\left(\int_{0}^{1}{e}^{x+t}{g(t)}dt\right)+x$$ I have tried to solve it by applying Newton-Leibnitz formula and solving the linear differential ...
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3answers
184 views

solution to a general integral $\int_0^\infty \frac{\cos(tx)}{x^2+k^2}e^{-sx}dx$

I would like to find a general solution to the integral: $$I(s,t,k)=\int_0^\infty \frac{\cos(tx)}{x^2+k^2}e^{-sx}dx$$ so far using the substitution $u=\frac xk$ I have managed to reduce this to: $$I(s,...
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1answer
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How to differentiate $f(a(y),y)$ with respect to $y$?

How to differentiate $f(a(y),y)$ with respect to $y$? What rule do we need to apply on it? Actually, I'm trying to understand the Leibniz’s Rule. $$\displaystyle\frac{d}{d_y}\int_{a(y)}^{b(y)}f(x,y)dx=...