Articles of integration

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is ‘supposed’ to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ f(x)\, dx=\int_{-a}^a \frac{d^n}{dx^n}\delta(x) f(x)\, dx =\int_{-a}^a(-1)^n\delta(x) \frac{d^nf}{dx^n}\, dx=(-1)^nf^{(n)}(0).$$ This argument seems like a hack and I have no idea what is actually going on. I want to write […]

Gamma Function Converges

I’m reading on pg 160 of Complex Analysis by Stein, and I am having trouble understanding the argument below — it is intuitively plausible, but I don’t see it rigorously. For s > 0, $\Gamma(s)=\int_0^\infty e^{-t}t^{s-1}dt$ The integral converges for each positive s because near $t = 0$ the function $t^{s-1}$ is integrable, and for […]

$\iiint e^{-x^2-2y^2-3z^2}dV$

I was given this question in class but I don’t understand how to do it. Evaluate the triple integral in $\mathbb{R}^3$ of $\iiint e^{-x^2-2y^2-3z^2}dV$. The hint was to use the idea that $\int e^{-x^2}dx = \sqrt \pi$, which I understand, but I don’t get how to apply it here… If anyone can help I would […]

Showing that $\int_x^{\infty}\frac{1}{u^2}e^{(-u^2/2)}du=\frac{1}{x}e^{(-x^2/2)}-\int_x^{\infty}e^{{(-u^2/2)}}du$

My texbook claims that integration by parts of the integral $\int_x^{\infty}\frac{1}{u^2}e^{\frac{-u^2}{2}}du$, with the hint that $d(-\frac{1}{u})=\frac{du}{u^2}$, gives $$\frac{1}{x}e^{\frac{-x^2}{2}}-\int_x^{\infty}e^{\frac{-u^2}{2}}du$$ Now when I try to express $\int sdv$ as $sv-\int vds$. I get $s=-\frac{1}{u}$, $ds=\frac{du}{u^2}$ then $$dv=\int-\frac{1}{u}e^{\frac{-u^2}{2}}du$$ which i cannot integrate. Whats wrong here? Thanks in advance!

Solving this complicated integral using the Residue Theorem

The following is an integral I am trying to evaluate $$I= \int_{-\infty}^\infty f(s) \, ds = \int_{-\infty}^\infty \frac{\frac{1}{(1- \ \ 2 \pi j s )^{m}}-1}{2\pi j s }\ e^{-2\pi j s \ \theta}\ ds $$ where $\theta$ is non-negative constant and $m$ is an positive integer. Someone helped me by saying that, I can solve […]

About a solution of Measure Theory and Integration

The problem is from Folland’s book of Measure Theory and Integration. In this problem, $(X,\mathcal{M}, \mu)$ is the measure space and $L^+$ is the space of measurable functions $f:X\to[0,\infty]$. The problem and the solution are showed below. I have just a little question about this solution. In this context, when we are talking about simple […]

Differentiating under integral for convolution

I have a function $f\in L^1(\mathbb{R}) $ and $g(x)=\dfrac{1}{2\sqrt{\pi t}}e^{-\frac{(at+x)^2}{4t}}$, where $a,t\in\mathbb{R}$, $t>0$. I want to show that $$\dfrac{d}{dx}\int_{-\infty}^\infty f(y)g(x-y)dy=\int_{-\infty}^\infty f(y)\dfrac{d}{dx}g(x-y)$$ Leibniz doesn’t work since there’s no continuity assumption on $f$. I’m thinking about using the dominated convergence theorem, but how would the proof go?

Help with integrating 101: $\int y \ln{y}\,\mathbb dy$

Would appreciate it if someone would please help me solve this $$\int y\;\ln y\, \mathbb dy$$ taking time to explain reason for each step taken. Thanks in advance!

Proving that there exists a unique f(x) given Area and arc-length of f(x) on a given interval

I’ve been suggested to this site by some nice people at mathoverflow.net Before I get started, let me tell you a little about myself. I’m a fourth year Mechanical Engineering student at the University of Michigan, Dearborn Campus. This said, I am not a mathematician, however I do know a bit about math, and I […]

Are there ways of telling whether or not an integral will have a closed expression?

I was looking at some of the great integral posts that have graced this website and I am wondering if there is a way to tell whether or not a definite integral will have a closed form. If there is no “one-size-fits-all” criterion, are there any general rules? (e.g. integrals of polynomial functions will have […]