Let $m$ be a probability measure on $Z \subseteq \mathbb{R}^p$, so that $m(Z)=1$. Consider a locally bounded $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, with $X \subseteq \mathbb{R}^n$, $Y \subseteq \mathbb{R}^m$ compact, such that $f(\cdot, \cdot, z)$ is continuous, and $f(x,y,\cdot)$ is measurable. Also, $f$ is uniformly ($\forall (x,y,z) \in X \times Y […]

As interested in factorization of integers, I had the idea to define the following natural numbers : z(n) := [$\int_n^{n+1} x^x dx$] My questions : 1) PARI can easily calculate z(n) numerically, but for large n it takes quite a long time. Is there an efficient method to calculate z(n) ? (The integral needs to […]

Given beta distribution as: $$ \mathcal{B}(x;a,b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} x^{a-1} (1-x)^{b-1} $$ I am trying to show: $$ \int_0^1 x^{a-1} (1-x)^{b-1}\,dx = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} $$ which can be considered as the normalizing constant. Here is what I have done so far (using the hints at Bishop’s). Write $\Gamma(a)\Gamma(b)$ and change variable $t=y+x$ fixing $x$: \begin{align*} \newcommand{\ints}{\int_0^\infty\!\!\int_0^\infty} \Gamma(a)\Gamma(b) […]

Let $f \in L^1(\mathbb{R})$. Find $$ \lim_{n \rightarrow \infty} \int_{-\infty}^\infty f(x)\sin(nx) dx \,. $$ LDCT is a no go, as well as MCT and FL, which are really the only integration techniques we’ve developed thus far in the course. Integration by parts isn’t applicable either since we just have $f \in L^1$. I’ve tried splitting […]

How to integrate $\displaystyle \int_a^b t \cdot \sqrt{\frac{t – a}{b – t}} \,dt$? I literally have no idea how to integrate this integral. I’ve tried all basic methods, but it seems like a quite hard problem for a beginner.

I need to show that $$\int_0^1 \left( \frac{1}{\ln(1+x)} – \frac{1}{x} \right) \,dx$$ converges, given that $$\lim_{x\rightarrow0^+} \left( \frac{1}{\ln(1+x)} – \frac{1}{x} \right) = \frac{1}{2}$$ I’m not sure how to do this because my text gave a very brief treatment on applying the limit comparison test for the improper integral of the second kind. Here is what […]

Can an analytic expression be given for $$\int \sqrt{ax^2 + bx +c} \, dx$$ I think substitution doesn’t work in this case (I need to compute the integral $\int_0^t \ldots$).

Is it true that $\int_0^A dx \int_0^B dy f(x) f(y) = 2 \int_0^A dx \int_0^x dy f(x) f(y)$ ? If so, can this be proved?

Consider the following numerical integration method:$$\int_a^b \left( f'(x) \right)^2\ dx \approx h\cdot \sum_{i=0}^{n-1} \left(\frac{f(x_{i+1}) – f(x_i)}{h}\right)^2$$ Where $h=\frac{b-a}{n}$ and $x_k = a+kh$. You may assume the derivatives (in any order) are bounded. What is the error for this method? So basically I could subtract the right-hand-side from the left-hand-side but I don’t see how to […]

I’m trying to solve the following differential equation $ y”’ = y$ and given conditions: $ y(1) = 3$, $y'(1) = 2$ and $y”(1) = 1 $ I began by making it: $y”’ – y = 0$ But I’m uncertain of what to do after, or if I’m even headed in the right direction. Please […]

Intereting Posts

Jordan normal form and invertible matrix of generalized eigenvectors proof
Verification of integral over $\exp(\cos x + \sin x)$
What Is Exponentiation?
Series $\frac{1}{4}+\frac{1\cdot 3}{4\cdot 6}+\frac{1\cdot 3\cdot 5}{4\cdot 6\cdot 8}+\cdots$
The preorder of countable order types
Paradox as to Measure of Countable Dense Subsets?
Transforming an integral equation using taylor series
Lipschitz and uniform continuity
Homological categories in functional analysis
Covariance of Ornstein-Uhlenbeck process
Help find the MacLaurin series for $\frac{1}{e^x+1}$
Eigenvalue of an Euler product type operator?
Compact operators: why is the image of the unit ball only assumed to be relatively compact?
convergence of alternating series — weakening a hypothesis
Is there much of difference between set models and class models?