# Questions tagged [leibniz-integral-rule]

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

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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
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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 ...
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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 ...
64 views

### 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 ...
37 views

### 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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### 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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### 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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1 vote
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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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### 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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1 vote
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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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