Articles of integration

Continuity of $\max$ of Lebesgue integral

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 […]

Calculation of special natural numbers

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 […]

Problem deriving Beta distribution normalizing constant

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) […]

Compute the limit of $\int_\mathbb{R} f(x)\sin (nx)$ when $n\to\infty$, for $f \in L^1$

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 $\int_a^b t \cdot \sqrt{\frac{t – a}{b – t}} \,dt$?

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.

How to show $\int_0^1 \left( \frac{1}{\ln(1+x)} – \frac{1}{x} \right) \,dx$ converges?

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 […]

Analytic expression for the primitive of square root of a quadratic

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 $\int_0^t \ldots$).

Proof for an integral identity

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?

Evaluate the error for a numerical integration custom method

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 […]

How to solve $y''' = y$

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 […]