Questions tagged [leibniz-integral-rule]
Also known as Feynman's trick or differentiation under the integral sign.
155
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How to solve the following limit
If
$$\lim_{x\to0}\frac1{x^m}\prod_{k=1}^n \int_0^x\big[k-\cos(kt)\big]\mathrm dt$$
exists and is equal to $20$ (where $m,n\in\mathbb N$) then what is the value of $n$?
I started this question with ...
- 250
2
votes
2
answers
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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 ...
- 125
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votes
0
answers
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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 ...
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votes
1
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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 ...
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1
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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 ...
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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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votes
1
answer
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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)}...
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2
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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 ...
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2
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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
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votes
1
answer
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Differentiation under the integral sign in higher dimensions
Let's say I have a function $f : \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n$. Are there conditions under which the following holds
$$ \frac{\partial}{\partial\mathbf{x}^T}\int_\Omega f(\mathbf{x},...
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1
answer
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Boundary velocity of moving surface in $\mathbb R^3$
Let's have a surface $\Sigma(t)$ moving in the three-dimensional space $\mathbb{R}^3$, with boundary $\partial \Sigma$ as in the below figure:
sketch-image
Let's have a space-dependent scalar $c(\...
1
vote
2
answers
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Propagation of error for an integral using Leibnitz' rule
I'm computing integrals $u_i$, where
$$u_i = \int_a^b s^i G(s) ds$$
and $a$ and $b$ are constants.
Now I want to estimate the error in $u_i$; I have estimates for the error $\sigma_G$ of $G$, and ...
- 133
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0
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48
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Leibniz rule for Lebesgue-Stieltjes integral
I am struggling to find whether there exists something like the Leibniz rule (interchanging integral and derivative) for Lebesgue-Stieltjes integrals.
Does such a result exist? If so, how to use it?
...
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1
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How does Leibniz' rule work in this equation
I am reading through the book Probability and Random Processes for Electrical and Computer Engineers . On page 307, there is an equation which states the following.
With $F_{X|Y}$ representing the CDF ...
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Generalized Leibniz's Rule
Let $\Omega$ be a open set of $\mathbb{R}^N$, $u: \Omega \to \mathbb{R}$ a function, $ u \in C^2(\Omega)$. Given $x_0 \in \Omega$, let $r > 0$ be such that $B_r(x_0) \subset \subset \Omega$. For $...
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First mean value theorem for definite integrals and differentiation
From the First MVT for integral we have ($g$ with constant sign) $\int_{a}^{b}f(x)g(x)\,dx=f(c)\int _{a}^{b}g(x)\,dx.$.
My question: is there a theorem that tells how to sign $\frac{dc}{db}$. Is the ...
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1
answer
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For which functions $g$ is $\displaystyle f(x) :=\int_{0}^{x}\int_{1}^{x-s}\frac{g(t)}{t}\space dt\space ds$ differentiable?
Differentiating $\displaystyle f(x)=\int_{0}^{x}\int_{1}^{x-s}\frac{g(t)}{t}\space dt\space ds$, I obtained $\displaystyle \int_{1}^{0}\frac{g(t)}{t}\space dt + \int_{0}^{x} \frac{\partial}{\partial x}...
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Leibniz type rule for the fourier based fractional derivative
Given $f(t) \in L^2(\mathbb{R})$, $\alpha \in \mathbb{R}$ and the definition
\begin{align}
D^\alpha f(t) = \mathcal{F}^{-1}\left( (i\omega)^\alpha \mathcal{F}\left(f(t);\omega\right);t\right)
\end{...
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Leibnitz's Rule regarding constants
I am going over Leibnitz's Rule, and unfortunately one part isn't making sense to me. Leibnitz's Rule states $$\frac {\partial}{\partial{y}}\int M(t, y)dt = \int \frac {\partial{M(t, y)}}{\partial{y}}...
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vote
1
answer
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Impossible counterexample to Leibniz integration rule
I am greatly confused about the Leibniz integral rule for probability densities. If $f_X$ is a valid univariate probability distribution parametrized by $\boldsymbol \theta=\{\theta_1,...,\theta_k\}$, ...
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1
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Under which assumptions does Leibniz integral rule hold for measureable subsets?
Suppose $f$ is some measurable function. The Lebesgue version of the Leibniz integral rule is, according to Wikipedia $$\frac{d}{dx}\int_{\Omega}f(x,y)dy=\int_{\Omega}\frac{\partial}{\partial x} f(x,y)...
- 136
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votes
0
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23
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Leibniz integral rule for finding integral derivative
I am wondering if my approach is acceptable. I am trying to compute the following with the Leibniz integral rule:
$\frac{d}{dt}\left(\int_{a(t)}^{b(t)}\,x'(t)^2\,x(t)^2\,dx\right)=[(\,x'(t)^2\,x(t)^2)\...
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0
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Leibniz rule on the boundary
I am trying to check for the Leibniz rule (measure theory version) holds for the boundary. Precisely, it states that
Let $I \subset \mathbb{R}$ be an open interval, $\Omega$ be a measure space and $\...
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How can I solve the integral equation $\pi a^2 (f(x))^2 = \int_{0}^{f(x)/a} \sqrt{(f(x))^2 -a^2t^2} \ \mathrm dt $ for the function $f(x)$?
How can I solve the integral equation $\pi a^2 (f(x))^2 = \int_{0}^{f(x)/a} \sqrt{(f(x))^2 -a^2t^2} \ \mathrm dt $ for the function $f(x)$, where $a$ is a constant?
I have tried turning this into a ...
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0
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Is there a generalization of Leibniz integral rule to contour integrals?
I am interested in applying Leibniz integral rule to the following general contour integral.
$$ g(r)=\int_{f(x,y)=r} h(x,y) d\Gamma(x,y) $$
Note that the dependence on r exists only through the ...
- 321
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1
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Applying Reynold's Transport Theorem on expanding sphere to differentiate under the integral sign with varying limit
I'm working through the proof of the mean value inequality (1.15) of Colding-Minicozzi's A Course on Minimal Surfaces, and I'm stuck on this subproblem. Let $\Sigma$ be a $k$-dimensional minimal ...
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1
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101
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Using Feynman's technique to find $\int_0^{\infty}\frac{\cos{x}}{x^2+1} dx$ [duplicate]
Using Feynman's technique, I want to evaluate $$\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}}{...
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0
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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 ...
- 1
5
votes
2
answers
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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 ...
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vote
1
answer
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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.
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1
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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 ...
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vote
1
answer
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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
$$
E(t) := \frac{1}{2}\int_{B_t} |\...
- 734
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0
answers
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Differentiating a standard normal cumulative distribution function [duplicate]
I am trying to find a way to differentiate the standard normal cumulative distribution function. I think it has something to do with the Leibniz rule but I can't wrap my head around it.
I would like ...
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votes
2
answers
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Interchanging a limit and a parametric improper integral
Suppose I have the limit:
$$\lim _{t\to 0^+}\int _0^{\sqrt{\sqrt{t}+4}}\sqrt{1+\frac{t}{4\sqrt{x}}}\:dx$$
How can I prove that the limit is $2$? It is easy to prove that for all $t > 0$ the ...
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votes
1
answer
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Condition on applying Leibniz rule to find min of $\int_0^{2\pi} (\sin x-ax-b)^2 dx $
I want find real $a$, $b$ such that $$F(a,b)=\int_0^{2\pi} (\sin x-ax-b)^2 dx$$ has minimun. Let $f(x,a,b)=(\sin x-ax-b)^2$.
I know I can directly find compute the integral in terms of $a$, $b$, then ...
- 126
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vote
0
answers
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Derivative of $\underset{x_1^2+...+x_m^2\leq r^2}{\int ...\int} f(x_1,...,x_m) dx_1 \, dx_2 ... dx_m$
I want to calculate the derivative
$$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$$
with respect to $r$. $f:\mathbb{R}^m\rightarrow \mathbb{R}$ is a ...
- 1,574
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vote
1
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Differentiate the following integral
Let the following integral
$$I(y)=2\int_{\sqrt {\frac 1 y -1}}^{\infty} \frac 1 {\pi} \frac 1 {1+t^{2}} dt.$$
Compute $\frac{dI}{dy}$
I'm trying to use Leibniz's formula but the limit $\infty$ ...
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votes
0
answers
49
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Leibniz integral rule with the measure CDF
Let $F$ be the cdf of a distribution. The distribution is not continuous and does not has a density function. Let $$g(x) = \int_{a(x)}^{b(x)}h(x, u)dF(u),$$ with $a$, $b$, $h$ continuous functions. As ...
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evaluate this definited integral using the differentiating under the integral sign Leibniz rule.q
I would like to solve this integral using differentiating under the integral sign or leibniz rule.
$$ \int_{0}^{\infty}\frac{e^{-ax}\sin(rx)}x\,\mathrm dx$$
Please let me know the steps to solve it.
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0
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Why does I'(t) output different values you set t=0 at different parts of the equation
So I was watching this video by Michael Penn https://www.youtube.com/watch?v=nkaZEI_e2SU&list=LL&index=8&t=430s and at around the 16:30 mark he puts in t = 0 into I'(t) and gets $-\pi$ as ...
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vote
2
answers
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Using Leibniz's Rule to Evaluate Integrals
Problem Evaluate the integral
$$\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{\...
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answer
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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 ...
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votes
1
answer
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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:
$$ \int_a^b \frac{\partial}{\partial y} f(x,y) \ dx = \frac{d}{dy} \left(\...
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vote
1
answer
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Verify that a certain function satisfy an ODE
The question is:
Show that the function
$y={2\over\pi}\int_0^{\pi\over2}\cos(x\sin(\theta))d\theta$
satisfy the ODE $y''+{y'\over x}+y=0$.
I tried use the Leibniz rule for integration and end up with ...
- 55
1
vote
1
answer
114
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Solving an integral using differentiation under the integral sign
$$\int\limits_0^\infty \exp\Big(-x^2-\frac{a^2}{x^2}\Big)\,\mathrm dx = \frac{\sqrt{\pi}}2e^{-2|a|}$$
How do we prove the above result? I tried using the Leibniz Integral rule but I ended up getting ...
0
votes
1
answer
112
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Differentiating an integral with variable as a limit
I want to differentiate the following function with respect to $x$: $$G(x) = \int_{0}^{x} (x-y)^nf(y)dy$$
I know how to solve the integral using the Leibniz rule, it is in the exact form which the ...
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votes
1
answer
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How does $\int_0^x\int_0^x...\int_0^x(x-t)u(t)dtdt...dt=\frac{1}{n!}\int_0^x(x-t)^nu(t)dt$?
We were giving the following equation to reduce multiple integrals ($n$-integrals) to single integrals
$$
\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
$$
However, ...
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vote
1
answer
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Special case of Leibniz formula
I am wondering if I understand it right. Considering the following partial derivative:
$$
\partial_tF(t, x) = \partial_t\int_{a(t)}^{b(t)} f(t-s, x)ds
$$
It looks to me as Leibniz should be used. So ...
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votes
1
answer
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Compute derivative of $k(t)=\int^{\infty}_{-\infty} \frac{\sin{tx^2}}{1+x^4}\,\textrm dx$
I'm trying to compute the derivative of $k(t) = \int^{\infty}_{-\infty} \frac{\sin{tx^2}}{1+x^4}\,\textrm dx$.
I've already showed that it exists, so here's what I've thought of so far:
letting $k_n(t)...
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1
answer
47
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Leibniz Rule with integral that has an integral in the integrand
I have the equation:
$$\ 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 ...
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