Questions tagged [leibniz-integral-rule]

Also known as differentiation under the integral sign.

Filter by
Sorted by
Tagged with
0 votes
1 answer
63 views

Using Feynman's technique to find $\int_0^{\infty}\frac{\cos{x}}{x^2+1} dx$ [duplicate]

Using Feynman's technique, I want to evaluate $$\int_0^{\infty}\frac{\cos{x}}{x^2+1} dx$$ I set $$I(a)=\int_0^{\infty}\frac{\cos{ax}}{x^2+1} dx$$ which gives $$I'(a)=\int_0^{\infty}\frac{-x\sin{ax}}{...
user avatar
  • 1,073
0 votes
0 answers
25 views

can we interchange the order of integration and differentiation in the following situation?

I want to move the partial differentiation inside of integration? is it legal mathmatically? namely from $\frac{\partial^2}{\partial x^2} \int_{0}^{\infty } \int_{0}^{2\pi } exp(2 \pi i txcos \theta)d ...
user avatar
  • 1
5 votes
2 answers
79 views

Interchanging spatial Fourier transform and time derivative for heat kernel

Let $K_t := (4\pi t)^{-n / 2}e^{|x|^2 / 4t}$ for $x \in \mathbb{R}^n$ and $t \in (0, \infty)$. I would like to show that $$ \tag{1} \partial_t \widehat{K_t} = \widehat{\partial_t K_t}, $$ (which makes ...
user avatar
  • 958
1 vote
1 answer
18 views

second fundamental form: unknown derivation

I'm unable to compute the second line in the proof double dot $r=r_{uu}\dot{u}^2+2r_{uv}...$ I think that I should use the chain rule together with the Leibnitz rule, but I do not know how.
user avatar
  • 3,578
2 votes
1 answer
66 views

Unsure where I've gone wrong with: $ \int_0^\infty \frac{1}{x}\sin\left(ax\right)\cos\left(bx^2\right)\:dx$

I've read over my working many times now and have been unable to resolve where my error lies. Can anyone please have a look and advise. Thank you! Note, in the following, the Cosine and Sine integrals ...
user avatar
  • 23
1 vote
1 answer
29 views

Uniqueness of Homogeneous Wave Equation with Initial Conditions

In Selberg's PDE lecture notes, to prove Theorem 1 (which is equivalent to proving uniqueness of homogeneous wave equation), he defined the energy function as $$ E(t) := \frac{1}{2}\int_{B_t} |\...
user avatar
  • 660
1 vote
0 answers
18 views

Differentiating a standard normal cumulative distribution function [duplicate]

I am trying to find a way to differentiate the standard normal cumulative distribution function. I think it has something to do with the Leibniz rule but I can't wrap my head around it. I would like ...
user avatar
4 votes
2 answers
100 views

Interchanging a limit and a parametric improper integral

Suppose I have the limit: $$\lim _{t\to 0^+}\int _0^{\sqrt{\sqrt{t}+4}}\sqrt{1+\frac{t}{4\sqrt{x}}}\:dx$$ How can I prove that the limit is $2$? It is easy to prove that for all $t > 0$ the ...
user avatar
  • 151
2 votes
1 answer
38 views

Condition on applying Leibniz rule to find min of $\int_0^{2\pi} (\sin x-ax-b)^2 dx $

I want find real $a$, $b$ such that $$F(a,b)=\int_0^{2\pi} (\sin x-ax-b)^2 dx$$ has minimun. Let $f(x,a,b)=(\sin x-ax-b)^2$. I know I can directly find compute the integral in terms of $a$, $b$, then ...
user avatar
1 vote
0 answers
70 views

Derivative of $\underset{x_1^2+...+x_m^2\leq r^2}{\int ...\int} f(x_1,...,x_m) dx_1 \, dx_2 ... dx_m$

I want to calculate the derivative $$F(r)=\underset{x_1^2+...+x_m^2\leq r^2}{\int ... \int} f(x_1,...,x_m) \, dx_1 \, dx_2 ... dx_m$$ with respect to $r$. $f:\mathbb{R}^m\rightarrow \mathbb{R}$ is a ...
user avatar
  • 1,406
1 vote
0 answers
41 views

Problem in the derivative with multi-index notation

I'm currently having some problems in understanding what our professor wrote. Text reads: $$\partial^{\alpha}\left((x-x_0)^{\alpha} \phi\left(\frac{x-x_0}{\epsilon}\right)\right) = \alpha!\phi\left(\...
user avatar
1 vote
1 answer
68 views

Differentiate the following integral

Let the following integral $$I(y)=2\int_{\sqrt {\frac 1 y -1}}^{\infty} \frac 1 {\pi} \frac 1 {1+t^{2}} dt.$$ Compute $\frac{dI}{dy}$ I'm trying to use Leibniz's formula but the limit $\infty$ ...
user avatar
0 votes
0 answers
31 views

Leibniz integral rule with the measure CDF

Let $F$ be the cdf of a distribution. The distribution is not continuous and does not has a density function. Let $$g(x) = \int_{a(x)}^{b(x)}h(x, u)dF(u),$$ with $a$, $b$, $h$ continuous functions. As ...
user avatar
  • 291
0 votes
1 answer
76 views

evaluate this definited integral using the differentiating under the integral sign Leibniz rule.q

I would like to solve this integral using differentiating under the integral sign or leibniz rule. $$ \int_{0}^{\infty}\frac{e^{-ax}\sin(rx)}x\,\mathrm dx$$ Please let me know the steps to solve it.
user avatar
0 votes
0 answers
25 views

Serie the taylor.

I trying to find the Taylor expansion around to $\varepsilon = 0$ of: $$\int_{0}^{\alpha_{1}(t,\varepsilon)}\left[ \varepsilon F_{1}^{0}(s,\varphi_{0}(s,x,\epsilon)) + \varepsilon^{2} F_{2}^{0}(s,\...
user avatar
  • 115
0 votes
0 answers
35 views

Why does I'(t) output different values you set t=0 at different parts of the equation

So I was watching this video by Michael Penn https://www.youtube.com/watch?v=nkaZEI_e2SU&list=LL&index=8&t=430s and at around the 16:30 mark he puts in t = 0 into I'(t) and gets $-\pi$ as ...
user avatar
  • 11
1 vote
2 answers
75 views

Using Leibniz's Rule to Evaluate Integrals

Problem Evaluate the integral $$\int_0^1 \frac{\ln(x+1)}{x^2+1} \, dx$$ using Leibniz's rule. Solution attempt. To evaluate the above integral, we consider a similar integral $$F(y) = \int_0^1 \frac{\...
user avatar
2 votes
1 answer
94 views

How do I show the function $I:\mathbb{R}^{+}\rightarrow\mathbb{R}$ defined by $I(x)=\int_{0}^{x} \frac{dt}{\sqrt{e^{x}-e^{t}}}$ has a unique maximum?

I've seen that $I$ satisfies that $$\lim_{x\to 0^{+}}I(x)=0$$and also that$$\lim_{x\to+\infty}I(x)=\lim_{x\to\infty}\frac{1}{\sqrt{e^{x}}}\int_{0}^{x}\frac{dt}{\sqrt{1-\frac{e^{t}}{e^{x}}}}=0,$$ so by ...
user avatar
  • 33
1 vote
0 answers
58 views

Swapping derivatives in a double integral expression with a variable limit

I want to confirm that my understanding of the following is correct. I'm given an expression of the form $$ \frac{\partial }{\partial x} \int_{b(x,t)}^{a(x,t)} \left(\frac{\partial}{\partial t} \int_z^...
user avatar
0 votes
1 answer
60 views

Applying Leibniz Integral Rule to Constant Limits of Integration

Using the Leibniz integral rule given here, Leibniz, it seems that for any arbitrary multivariable function $f(x,y)$, we have: $$ \int_a^b \frac{\partial}{\partial y} f(x,y) \ dx = \frac{d}{dy} \left(\...
user avatar
  • 868
0 votes
0 answers
21 views

Partial derivative of a heat kernel

I happen to have the heat kernel on the two-dimensional hyperbolic space and I need to take partial derivatives in order to check that it satisfies the heat equation as expected. The problem is I can ...
user avatar
1 vote
1 answer
45 views

Verify that a certain function satisfy an ODE

The question is: Show that the function $y={2\over\pi}\int_0^{\pi\over2}\cos(x\sin(\theta))d\theta$ satisfy the ODE $y''+{y'\over x}+y=0$. I tried use the Leibniz rule for integration and end up with ...
user avatar
0 votes
0 answers
27 views

differentiation under the integral sign and jump discontinuity

I have the following integral: $$ \frac{d}{d\mathbf{t}}\int_{E}g(\mathbf{t},\mathbf{x})f(\mathbf{x})d\mathbf{x}, $$ where $\mathbf{t}\in\mathbb{R}^{k}$, $E$ is a compact subset of $\mathbb{R}^{m}$, $g:...
user avatar
  • 35
1 vote
1 answer
94 views

Solving an integral using differentiation under the integral sign

$$\int\limits_0^\infty \exp\Big(-x^2-\frac{a^2}{x^2}\Big)\,\mathrm dx = \frac{\sqrt{\pi}}2e^{-2|a|}$$ How do we prove the above result? I tried using the Leibniz Integral rule but I ended up getting ...
user avatar
0 votes
1 answer
85 views

Differentiating an integral with variable as a limit

I want to differentiate the following function with respect to $x$: $$G(x) = \int_{0}^{x} (x-y)^nf(y)dy$$ I know how to solve the integral using the Leibniz rule, it is in the exact form which the ...
user avatar
2 votes
1 answer
66 views

How does $\int_0^x\int_0^x...\int_0^x(x-t)u(t)dtdt...dt=\frac{1}{n!}\int_0^x(x-t)^nu(t)dt$?

We were giving the following equation to reduce multiple integrals ($n$-integrals) to single integrals $$ \int_0^x\int_0^x...\int_0^x(x-t)u(t)dtdt...dt=\frac{1}{n!}\int_0^x(x-t)^nu(t)dt $$ However, ...
user avatar
  • 1,811
1 vote
1 answer
80 views

Special case of Leibniz formula

I am wondering if I understand it right. Considering the following partial derivative: $$ \partial_tF(t, x) = \partial_t\int_{a(t)}^{b(t)} f(t-s, x)ds $$ It looks to me as Leibniz should be used. So ...
user avatar
2 votes
1 answer
68 views

Compute derivative of $k(t)=\int^{\infty}_{-\infty} \frac{\sin{tx^2}}{1+x^4}\,\textrm dx$

I'm trying to compute the derivative of $k(t) = \int^{\infty}_{-\infty} \frac{\sin{tx^2}}{1+x^4}\,\textrm dx$. I've already showed that it exists, so here's what I've thought of so far: letting $k_n(t)...
user avatar
0 votes
0 answers
41 views

How to derive the Leibniz rule for the Grassmann number?

Consider the Grassmann number $$\{\theta_i,\theta_j\}=0$$. The Leibniz rule for the Grassmann number was given as definition, $$\frac{d\theta_i\theta_j}{d\theta}=\frac{d\theta_i}{d\theta}\theta_j-\...
user avatar
0 votes
1 answer
45 views

Leibniz Rule with integral that has an integral in the integrand

I have the equation: $$\ V_t = \int_t^T e^{\int_t^s r_x dx} X_sds $$ I need to take the derivative of this with respect to $t$. This seems like an obvious Leibniz rule problem, but I am unsure how to ...
user avatar
  • 37
0 votes
1 answer
95 views

$f\left(x\right) = \int_{1}^{x}\frac{\sin(x)\cos(y)\mathrm{d}y}{y^{2}+y+1}$, find solution of $f'\left(x\right)=0$

My Approach: Using Leibniz Rule: $f'\left(x\right)=\frac{\sin x\cos x}{x^{2}+x+1}=\frac{\sin2x}{2\left(x^{2}+x+1\right)}$. Now $f'\left(x\right)=0$ when $\sin2x=0\to2x=n\pi\to x=\frac{n\pi}{2}$ So ...
user avatar
  • 923
0 votes
1 answer
27 views

Is there some rule to derive a sum of $x$, with lower and upper bounds depending on $x$?

I like to know a rule to derivate this: $$ g(x) = \sum_{t=a(x)}^{b(x)} f(x,t)$$ $$ a(x) < b(x) $$ $$ a(x), b(x) \in \mathbb{Z} $$ I already tried the chain rule (like in the Leibniz integral rule): ...
user avatar
2 votes
0 answers
47 views

Strong Version of Leibnitz Rule for Differentiation under the Expectation

The Leibnitz rule for differentiation under the expectation provides conditions such on a function $g: \mathbb{R} \times \mathcal{X} \to \mathbb{R}$ such that \begin{align} \frac{d}{dt} E[g(t,X) ] \...
user avatar
  • 5,385
0 votes
0 answers
47 views

Exchange Partial Derivative wrt. to a Function (Function is depedent on variable of integration)

At the moment I am facing a challenge concerning the exchange of an integral and a partial derivative. Not sure, if it is right, but is there any assumption that the following holds or what is the ...
user avatar
1 vote
2 answers
119 views

Why $f(x)$ cannot be obtained clearly from $\int ^{b}_{a} (f(x)-3x)\; dx=a^2-b^2$

If $$\int ^{b}_{a} (f(x)-3x)\; dx=a^2-b^2$$ then the value of $f(\frac{\pi}{6})$ is Ans: [$\frac{\pi}{6}$] OR [$\frac{\pi}{6},\frac{2\pi}{3},\frac{\pi}{3},\frac{\pi}{2},\pi$ {ie all given options}] ...
user avatar
  • 295
0 votes
0 answers
22 views

Deriving a differential equation for t between a function and it's time derivative

In a previous part of the problem, Given the function $g(t)=\int_0^\infty e^{\frac{x^2}{2}}cos(tx)dx$ , $\frac{dg(t)}{dt}$ is derived using Leibniz Rule to be: $$\frac{dg(t)}{dt}=\int_0^\infty -xe^{\...
user avatar
2 votes
0 answers
129 views

Evaluating $\frac{1}{2\pi i}\int^{a+i\infty}_{a-i\infty}\frac{x^s}{s-\beta}ds$ using Feynman integration

I am trying to prove that $$ \frac{1}{2\pi i}\int^{a+i\infty}_{a-i\infty}\frac{x^s}{s-\beta}ds =\begin{cases} x^{\beta}, & x > 1 \\ 0, & 0 < x < 1 \\ \end{cases} $$ for $0<{\rm Re}(...
user avatar
2 votes
1 answer
162 views

Evaluating $\int_{0}^{\infty}e^{-x^2}\:dx$, I get a wrong answer: $-\frac34\sqrt{\pi}$. Where is my mistake?

I did this calculation and something just isn't quite right. I am getting the result $$\int_{0}^{\infty}e^{-x^2}\:dx=\frac{-3\sqrt{\pi}}{4}$$ which is absolutely wrong. I would be grateful if anyone ...
user avatar
1 vote
0 answers
39 views

Conditions for differentiating under integral sign with mixed finite and infinite limits

I am working through Problem 2.18 of Statistical Inference by Casella & Berger. It is asking to prove that, for $X$ a continuous random variable with median $m$, we have $$\min_a \textrm{E}|X-a|=\...
user avatar
0 votes
0 answers
23 views

Derivative of a Bivariate normal CDF with respect to its variables

Following up on the question (and answers) here, I'm trying to derive $\frac{\partial \Phi(x_1, x_2|\mathbf{\underline{\theta}})}{\partial x_1}$ and $\frac{\partial \Phi(x_1, x_2|\mathbf{\underline{\...
user avatar
  • 129
1 vote
2 answers
137 views

What is the derivative of $\int_0^x \frac{f(y)}{\sqrt{x-y} }$?

Suppose I have some integral $$\int_0^x \frac{f(y)}{\sqrt{x-y} }$$ How would I differentiate this with respect to $x$? The Leibniz rule reads $$\frac{d}{dx} \int_0^x g(x,y) \ dy = g(x,x) +\int_0^x \...
user avatar
0 votes
0 answers
34 views

Application of Leibniz Rule to Integral Differentiation

I have a question I'm struggling with for quite a few hours now. For most of you its probably kindergarten mathematics. The problem: Implicitly differentiate the following with respect to $x$ $$ y = ...
user avatar
0 votes
3 answers
136 views

$\int_0^1 e^{-x^2}dx$ a clever way to get an answer

So I am working on a problem, in that I need to integrate $\int_0^1 e^{-x^2}dx$. So I did it in this way: $$\int_0^1 e^{-x^2} dx=\int_0^1 \sum_{n=0}^\infty \frac{(-x^2)^{n}}{n!} = \sum_{n=0}^\infty \...
user avatar
0 votes
1 answer
63 views

Continuity of the following integral.

Let $f(x,y):[0,1]^2 \rightarrow R$ be a continuos function, and $F(x)$ defined as $$F(x)= \int^{1}_{0} f(x,y)1_{(y\leq x)} dy.$$ where $1_{(y\leq x)}= 1$ if $y\leq x$ and $1_{(y\leq x)}=0$ if $y> x$...
user avatar
  • 1
0 votes
1 answer
27 views

How to find the derivative of an integral whose integrand is a composition of the integral limits?

Given a function $q=q(t, e)$, where $ e= e(t, v(\tau))$, such that $q = q(t, e(t, v(\tau)))$, how to find the expression for $\frac{dQ}{d \tau}$ if $$Q(\tau) = \int_0^\tau q(t, e(t, v(\tau))) dt$$ ...
user avatar
9 votes
2 answers
462 views

Feynman's trick to evaluate the integral $\int\limits_{0}^{2\pi}\sin^{8}(x)dx$ [closed]

I would like to evaluate the following integral using differentiation under the integral sign. $$\int\limits_{0}^{2\pi}\sin^{8}(x)dx$$ Unfortunately, I can't come up with a proper choice for a ...
user avatar
5 votes
1 answer
81 views

Does the limit $\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \sin \sqrt{t} d t}{x^{3}}$ exists?

Evaluate $$L=\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \sin \sqrt{t} d t}{x^{3}}$$ Since the limit is $\frac{0}{0}$ form, Using L'Hopital's rule and Leibniz rule we get $$L=\lim_{x \to 0}\frac{2x\...
user avatar
4 votes
1 answer
131 views

Applying Leibniz's Rule to an Integral With Multiple Parameters

I was given the following exercise and wasn't really able to make heads-or-tails out of it. It goes like so: Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be twice differentiable continuously, and ...
user avatar
0 votes
1 answer
74 views

How alternating series with sine and cosine fit definition of Alternating Series?

I was reading the definition of $\textit{Alternating Series}$, in wikipedia. https://en.wikipedia.org/wiki/Alternating_series All terms $a_n>0$ and the signs of the general terms alternate between ...
user avatar
  • 168
0 votes
1 answer
252 views

Leibniz rule for expected value

Let $Z(x)$ be a continuous random variable and let $f(x)$ be a probability density function defined on $: [0,\infty]$. Apply the Leibniz rule to the expected value $$E[Z(x)] = \int_{0}^\infty Z_i(x) f(...
user avatar
  • 155