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 evaluate the Gaussian integral $$ \int^\infty_0 e^{-x...

Where is differentiating under the integral sign useful and where is it used? What are some sources that explain this well? I'm doing some reading and a little confused.

$$\int_0^{\infty} \frac{\sin^2(x)}{x^2(x^2+1)} dx$$ The integral is equals with $\frac{\pi}{4}+\frac{\pi}{4e^2}$, but i can't prove it.

After reading articles on differentiation under the integral sign, I hit this post from MIT OCW, where after introducing the power tool, it challenges reader to do $$\int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)} dx$$ Obviously I have no clue where to start. Could any one give a hint?

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 found the right way yet. we have singularities a...

Consider $$F(z) = \int_0^\infty\frac{1-\cos(yz)}{y}\ {e^{-y}} dy$$ How do I use the Leibniz Rule to solve this? I've tried to follow the general rule but can't seem to get the answer which I am meant to prove, which is: $$\frac{dF}{dz} = \int_0^\infty \sin(yz){e^{-y}} dy$$ Thanks!

Leibniz rule: $$\frac{\partial}{\partial x}\int_{a}^{b}f\left(x,y\right)dy=\int_{a}^{b}\frac{\partial f\left(x,y\right)}{\partial x}dy$$ But what if $x$ is a function $y$? (Assume it's smooth and has any nice properties you want, but it isn't a constant.) Can I apply Leibniz rule and say $$\fr...

The Leibniz rule is as follows: $$\frac{d}{d\alpha} \int_{a(\alpha)}^{b(\alpha)} f(x, \alpha) dx = \frac{db(\alpha)}{d\alpha} f(b(\alpha), \alpha) - \frac{da(\alpha)}{d\alpha} f(a(\alpha), \alpha) + \int^{b(\alpha)}_{a(\alpha)} \frac{\partial}{\partial\alpha} f(x, \alpha) dx$$ What I would like...

Given the Leibniz rule: $$\frac{d}{dy}\int^a_b f(x,y) dx = \int_b^a \frac{\partial f}{\partial y} (x,y) dx$$ How do I prove a more general case using the chain rule and the above: $$\frac{d}{dy} \int_{g_1(y)}^{g_2(y)} f(x,y) dx =?$$ From the fundamental theorem of calculus we have that: $$\fr...

Let's say I have an integral: $$\int_{B_{t}} f(t,x) \mathrm{d}x$$ where $B_t$ is the ball of radius $t$. And I would like to apply the Leibniz rule to compute the derivative of this. How would I do it? I know that the formula goes something like this: $$\frac{d}{dt}\int_{B_{t}} f(t,x) \mathrm...

While trying to prove something, I came up with a strange case involving the use of Leibniz integral rule. Let a variable $x$ be parametrized by $t$ so that $x=x(t)$. Now I want to take the derivative with respect to $x(t)$ an integral: $$\frac{d}{dx(t)}\int_0^tf(x(t'))dt'$$ for a well-behave...

I have a feeling this post won't met the community guidelines (will delete if so). 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 ...

As part of going through a set of definite integrals that are solvable using the Feynman Trick, I am now solving the following: $$ \int_{0}^{\frac{\pi}{2}} \ln\left|2 + \tan^2(x) \right| \:dx $$ I'm seeking methods using the Feynman Trick (or any method for that matter) that can be used to sol...

I've been tracking this post today on math.SE where the OP was asking for a proof of convergence for the integral $$ \int_0^1 \frac{\ln\left(1-\alpha^2x^2 \right)}{\sqrt{1-x^2}}dx. $$ I want to ask a related but distinct question of how to explicitly compute the value of this integral. Somethi...

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).

How can I perform derivative of double integral $$\frac{\mathrm d}{\mathrm dt}\int_{t-\mathrm d1}^t \int_h^t f(s) \,\mathrm ds\,\mathrm dh$$ Can I apply a Leibniz rule of some form? How?

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 $$\frac{d}{dt}\int_{\phi(t)}^{\infty} f(t,s) ds ?$$

Suppose I need to compute the derivative $$ \frac{d}{dr} \int_{-\infty}^{\infty} \int_{h(r)}^\infty \int_{g(r)}^\infty {rf(x,y,z)\, dz\, dy\, dx}. $$ Can I apply a Leibniz rule of some form? How?

I am reading a paper that states: We note that if an integrable function satisfies the Lipschitz condition of order one, then differentiation and integration can be interchanged. This provides a more compact way to take the derivative. Consequently, in our proofs, if an integrable function s...

I have the following integral $$\int_a^b f(w, t)dt$$ where $w \in \Bbb R^n$ and I need to compute partial derivatives with respect to all components of $w$. How can I apply Leibniz rule to this problem? Suggested answer:

Why can't I apply Leibniz' rule in the following way? $$\frac{d}{ds} g(s)\int_0^\infty f(s,x,u) \, du = \int_0^\infty \frac{d}{ds}g(s)f(s,x,u)\,du,$$ assuming $gf$ and $(gf)'$ are continuous on $[0,+\infty]\times [s_0,s_1]$ for some $s_0<s_1\in\mathbb{R}$.

One can show using Leibniz' rule that $$\int_0^1 \frac{x^n-1}{\ln x} dx = \ln|n+1|.$$ To be specific, if we set $g(n) := \int_0^1 \frac{x^n-1}{\ln x} dx$, then \begin{align} g'(n) &= \frac d{dn} \int_0^1 \frac{x^n-1}{\ln x} dx \\ &= \int_0^1 \frac{\partial}{\partial n} \frac{x^n-1}{\ln x} \, dx...

I've been trying to find solve this integral for a while, using differentiation under the integral sign: $$\int_0^1 \! \frac{{e^{-ax}}\sin(x)}{x} \, \mathrm{d}x$$ But I keep getting stuck around here, when I'm trying to find the indefinite integral with respect to $x$: $$-a\int\!e^{-ax}\sin(x)...

I wish to know if the Leibniz rule is correctly applied in the following equation or I am missing something: $$\int \int \frac{\partial f(x,y,z)}{\partial x} dy dz =\frac{\partial \left(\int \int f(x,y,z)dy dz \right)}{\partial x} $$ $x, y$ and $z$ are independent of each other. The integrat...