Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [leibniz-integral-rule]

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

In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form $$ \int_{a(x)}^{b(x)}f(x,t)\, dt \text , $$ where $ -\infty < a(x),b(x)< \infty $, the derivative of this integral is expressible as $$\frac{d}{dx}\int_{a(x)}^{b(x)}f(x,t)\, dt=f\big(x,b(x)\big)b'(x)-f\big(x,a(x)\big)a'(x)+\int_{a(x)}^{b(x)}\frac{\partial}{\partial x}f(x,t)\,dt\text.$$ In particular,

$$\frac{d}{dx}\int_a^bf(x,t)\, dt=\int_{a}^{b}\frac{\partial}{\partial x}f(x,t)\,dt.$$

205 questions
52
votes
1 answer

Integrating $\int^{\infty}_0 e^{-x^2}\,dx$ using Feynman's parametrization trick

I stumbled upon this short article on last weekend, it introduces an integral trick that exploits differentiation under the integral sign. On its last page, the author, Mr. Anonymous, left several exercises without any hints, one of them is to…
Shuhao Cao
  • 19,689
43
votes
9 answers

Definite integrals solvable using the Feynman Trick

I'm looking for definite integrals that are solvable using the method of differentiation under the integral sign (also called the Feynman Trick) in order to practice using this technique. Does anyone know of any good ones to tackle?
user150203
41
votes
3 answers

Will moving differentiation from inside, to outside an integral, change the result?

I'm interested in the potential of such a technique. I got the idea from Moron's answer to this question, which uses the technique of differentiation under the integral. Now, I'd like to consider this integral: $$\int_{-\pi}^\pi \cos{(y(1-e^{i\cdot…
Matt Groff
  • 5,749
24
votes
2 answers

Differentiation under the integral sign for Lebesgue integrable derivative

The problem is the following: Let $a,b,c,d \in \mathbb R$ be given such that $a
Sam
  • 16,398
20
votes
2 answers

Differentiating Definite Integral

I think $\frac{d}{dx} \int f(x) dx = f(x)$ right? So $\frac{d}{dx} \int^b_a f(x) dx = [f(x)]^b_a = f(a)-f(b)$? But why when: $$f(x) = \int^{x^3}_{x^2} \sqrt{7+2e^{3t-3}}$$ then $$f'(x) = \color{red}{(x^3)'}\sqrt{7+2e^{3x-3}} -…
Jiew Meng
  • 4,663
19
votes
5 answers

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting integral and it should have a nice ODE. I have not…
Jeff Faraci
  • 10,393
18
votes
1 answer

Differentiating Under the Integral Proof

There are many variations of "differentiating under the integral sign" theorem; here is one: If $U$ is an open subset of $\mathbb{R}^n$ and $f:U \times [a,b] \rightarrow \mathbb{R}$ is continuous with continuous partial derivatives $\partial_1 f,…
ItsNotObvious
  • 11,263
12
votes
1 answer

Derivative of a double integral over a variable circular region

Calculate the following derivative $$\frac{d}{dt}\iint_{D_t}F(x,y,t) \, \mathrm d x \mathrm d y$$ where $$D_t = \lbrace (x,y) \mid (x-t)^2+(y-t)^2\leq r^2 \rbrace$$ I've read here$^1$ that the answer to the above question is in the form…
SMA.D
  • 1,497
11
votes
3 answers

how to solve $\int _0^1\frac{\ln \left(1+x\right)}{a^2+x^2}\:\mathrm{d}x$

how to solve $$\mathcal{J(a)}=\int _0^1\frac{\ln \left(1+x\right)}{a^2+x^2}\:\mathrm{d}x$$ i used the differentiation under the integral and got \begin{align} \mathcal{J(b)}&=\int _0^1\frac{\ln…
user815913
10
votes
2 answers

Evaluate integral $\int_0^\frac{\pi}{2} \ln\left(\frac{1+a\cos x}{1-a\cos x}\right) \frac{\mathrm{d} x}{\cos x}$ for $\left|a\right|<1$

Question: Evaluate the following definite integral: $$ I= \int_{0}^{\pi/2} \ln\left(1 + a\cos\left(x\right) \over 1 - a\cos\left(x\right)\right)\, {{\rm d}x \over \cos\left(x\right)}\qquad \mbox{where}\qquad\left\vert\,a\,\right\vert < 1 $$ This…
Frenzy Li
  • 3,774
10
votes
2 answers

Leibniz rule for improper integral

We know that the Leibniz integral formula $$\frac{d}{dt}\int_{\phi(t)}^{\psi(t)} f(t,s) ds = \int_{\phi(t)}^{\psi(t)} \frac{d}{dt}f(t,s) ds+f(t,\psi(t))\frac{d}{dt}\psi(t) -f(t,\phi(t))\frac{d}{dt}\phi(t).$$ Can we apply this rule for…
Worawit Tepsan
  • 686
9
votes
2 answers

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

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 function with a parameter. $\sin^{8}(ax)$ won't work…
GuntherOnFire
  • 183
9
votes
8 answers

Show that $\int_0^\pi \log( 1 - 2r\cos(t) + r^2)\, dt=0$

$\newcommand{\on}[1]{\operatorname{#1}}$ $$ \mbox{Show that for}\ r \in \left(-1,1\right),\ \int_{0}^{\pi}\log\left(1 - 2r\cos\left(t\right) + r^{2}\right){\rm d}t = 0 $$ Here's what I did so far: $$ \on{f}\left(r,t\right) = \log\left(1 -…
zartos
  • 125
9
votes
5 answers

Integrating $\int_0^\pi x^4\cos(nx)\,dx$ using the Feynman trick

I should solve the following integral $$\displaystyle\int_0^\pi x^4\cos(nx)\,dx$$ Usually you would integrate 4 times by parts. I was wondering if there is a more direct way, something like the Leibniz rule (aka Feynman trick).
Mathieu
  • 607
8
votes
1 answer

Partial derivative of integral: Leibniz rule?

The Leibniz rule is as follows: $$\frac{d}{d\alpha} \int_{a(\alpha)}^{b(\alpha)} f(x, \alpha) dx = \frac{db(\alpha)}{d\alpha} f(b(\alpha), \alpha) - \frac{da(\alpha)}{d\alpha} f(a(\alpha), \alpha) + \int^{b(\alpha)}_{a(\alpha)}…
Jase
  • 644
1
2 3
…
13 14