Questions tagged [leibniz-integral-rule]
Also known as Feynman's trick or differentiation under the integral sign.
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Question regarding integral involving logarithm and sine [duplicate]
I have to compute the following integral
$$\int_{0}^{\pi/2} \frac{\ln(1-\sin x)}{\sin x} dx$$
I decided to solve this using the Feynman's Trick for integration and parametrized the integral as follows
...
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Why this formula cannot be used here? [duplicate]
My book says
Why can't I apply it to
$$g(x)=\int_{0}^{x}e^{x-t}dt$$
Which gives $g'(x)=e^{x-x}$
but when we integrate properly we get $g'(x)=e^x$
What I thought
$g(x)=\int_{u(x)}^{v(x)}f(t,x)dt$
As ...
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Using Leibniz Integral Rule to Evaluate a log-sine Integral
I am looking for someone to check my work here.
$$I(a)=\int_0^{\frac{\pi}{2}}\ln(1+a\sin^2(x))dx$$
Using Leibniz/Feynman's trick, I differentiate both sides with respect to $a$, leaving me with
$$I'(a)...
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Derivative of Lebesgue integral with indicator functions
Suppose we have a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$.
I want to take the derivative of
$$
W\left(x\right)\equiv\int_{\Omega}\mathbf{1}\left\{ \omega\in R\left(x\right)\...
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Evaluate the derivative of the integral to get the specific form
$$
\frac{d}{dt}{\alpha}\left(t\right)=\frac{d}{dt}\left[2 \int_{t}^{\infty} \frac{d x}{x \sqrt{\left(\frac{x}{t}\right)^2\left(1-\frac{1}{t}\right)-\left(1-\frac{1}{x}\right)}}-\pi\right]
$$
$$
...
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Derivative of $ f(x)=\int_{1 / x}^{e^x / x} \frac{\cos (x t)}{t} d t $
Find the derivative of the function
$$
f(x)=\int_{1 / x}^{e^x / x} \frac{\cos (x t)}{t} d t, \quad(x>0) .
$$
I use $\frac{d}{dx}\int_{1 / x}^{e^x / x} \frac{\cos (x t)}{t} d t=\frac{x\cos(e^x)}{e^x}...
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Trying to solve $I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx$
While trying to find an answer to this problem on the forum, I came across this integral:
$$ I=\int_c^\infty {\sin \big(x+{k \over x} \big) \over x}dx \tag 1$$
Where $c$ and $k$ are real numbers. I ...
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Leibniz integral rule help
Let
$$I:=\frac{\partial}{\partial \epsilon} \left[\epsilon \int^{b(\epsilon)}_{a(\epsilon)} x f(x) dx \right]_{\epsilon = 0}.$$
My textbook claims that, as a consequence of the Leibniz integral rule:
$...
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Reasonable and Logical Integration
$\lim _{k\to +\infty}$ $\int_{0}^{k[x]} (kt-[kt])^k dt$ $;$ $k\in N$ is $[\frac{\lambda x} 2]$
Where $[.]$ denotes greatest Integer function then the value of $\lambda$ would be?
My approach to this ...
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A peculiar integral using Leibnitz rule of integration?
Mathworld while explaining (rather very briefly) the Leibnitz rule of integration, aka derivative under an integral sign, mentions that this method may be used to evaluate peculiar integrals such as $$...
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Variable limits (function of x) in a definite integral
I am stuck in a concept of Definite Integration.
Let us say we have a function that goes like ${f(x) = \int_0^x e^{x-t} f(t) \,dt}$
Now I wanted to know that if I put $x=0$, will the limit range from $...
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partial derivatives and integrals
suppose I have a function $$v(x,t):=\int_0^t u(x,t;s) ds.$$
Why $$v_t(x,t)=u(x,t;t)+\int_0^t u(x,t;s) ds.$$
I do not know where does the $u(x,t;t)$ term come from ?
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Evaluating $\int_1^{\sqrt2} \frac{\tanh^{-1}(\sqrt{2-x^2})}{1+x} dx$
I was trying to compute the value of the integral
$$\ I = \int_1^{\sqrt2} \frac{\operatorname{arctanh}(\sqrt{2-x^2})}{1+x}dx$$
I began by declaring the family of integrals:
$$\ I(a) = \int_1^{\sqrt2} \...
user1173615
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Two-dimensional generalization of Leibniz's integral rule
Given a function $f(x,y):\mathbb R\times \mathbb R\to \mathbb C$ and a real parameter $\theta$, one can use Leibniz's integral rule to solve
\begin{equation}\label{eq}\tag{1}\frac{d}{d\theta}\int_{a(\...
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How to Calculate the Partial Derivative F_u(1,v) of the Integral Function F(u,v)
I'm working on a problem in calculus and am having difficulty with a specific function and its partial derivative. The function is defined as:
$F(u,v) = \int_{uv}^{u+v}e^{-(u-y)^2}dy$
I'm trying to ...
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