# Questions tagged [leibniz-integral-rule]

Also known as differentiation under the integral sign.

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• 660
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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 ...
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 ...
• 151
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 ...
• 114
1 vote
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 ...
• 1,406
1 vote
41 views

• 115
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 ...
• 11
1 vote
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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 ...
• 37
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 ...
• 923
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### 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): ...
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) ] \...
• 5,385
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 ...
1 vote
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}] ...
• 295