Articles of integration

How to compute $\int \sqrt{\frac{x^2-3}{x^{11}}}dx$

How to compute the following integral ?? $\displaystyle\int \sqrt{\frac{x^2-3}{x^{11}}}dx$ I use the wolfram Alpha to get an ugly answer^2-3%29%2Fx^%2811%29%29^0.5+dx But there is no step for me

If the left Riemann sum of a function converges, is the function integrable?

If the left Riemann sum of a function over uniform partition converges, is the function integrable? To put the question more precisely, let me borrow a few definitions first. Pardon my use of potentially non-canon definitions of convergence. Given a bounded function $f:\left[a,b\right]\to\mathbb{R}$, A partition $P$ is a set $\{x_i\}_{i=0}^{n}\subset\left[a,b\right]$ satisfying $a=x_0\leq x_1\leq\cdots\leq x_n=b$. The […]

How to solve $ \displaystyle\int \bigg(\small\sqrt{\normalsize x+\small\sqrt{\normalsize x+\sqrt{ x+\sqrt{x}}}}\;\normalsize\bigg)\;dx$?

How to solve this: $$\displaystyle\int \bigg(\small\sqrt{\normalsize x +\small\sqrt{\normalsize x +\small\sqrt{\normalsize x +\sqrt{x}}}}\;\normalsize\bigg) \;dx$$

Finding values for integral $\iint_A \frac{dxdy}{|x|^p+|y|^q}$ converges

Given the following integral $$\iint_A \frac{dxdy}{|x|^p+|y|^q}$$ where $A=|x|+|y|>1$. How can one find for which $p$ and $q$ values the integral converges? Since the function is non-negative it is sufficient to show convergence/divergence on any Jordan exhaustion of $A$ in order to show convergence/divergence on $A$. I tried to use polar coordinates here, but I don’t […]

Leibniz rule, multiple integrals

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?

Change the order of integrals:$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$

$$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$$ From this it is obvious that $x\in[0,1],y\in[0,1-x],z\in[0,x+y]$. For it asks for the order to be in $$\int dz\int dx\int f(x,y,z)dy$$ . My method of doing this is for start from the outer variable ,in this case z, and from the original one can deduct that $z\in [0,1]$ But the professor goes on to make […]

Why $2n \int ^\infty _n \frac{c}{x^2 \log(x)} \sim n \frac{C}{n \log(n)}$?

I want to understand why we have $$ 2n \int ^\infty _n \frac{c}{x^2 \log(x)} \operatorname*{\sim}_{n\to\infty} n \frac{C}{n \log(n)} $$ where $c$ is a normalizing constant. I am unable to understand how the integral is removed.

$f(n) = \int_{1}^{n} n^{x^{-1}}dx$. $\frac{df}{dn}$?

Lets start with the domain $1 \leq n \in \mathbb{R}$: $f(n) = \int_{1}^{n} n^{x^{-1}}dx$ $\frac{df}{dn}$ should be sort of logarithmic and probably not interesting.? What is it, though? What have I tried? To see how far it is from something interesting, and maybe more..

evaluate the following integral

$$ \int J_0(x)\sin x~{\rm d}x $$ Where $J_0$ is Bessel function of first kind of order $0$ This what I tried $$ \int J_0(x)\sin x~{\rm d}x= -J_0(x) \cos x – \int J_0′(x)\cos x~{\rm d}x $$ $$ J_0′(x)=-J_1(x) $$ $$ \int J_0(x)\sin x ~{\rm}x= -J_0(x) \cos x -(J_1(x)\sin x – \int J_1′(x)\sin x~{\rm d}x) $$ $$ […]

Euler-Mascheroni constant: understanding why $\lim_{m\rightarrow \infty} \sum_{n=1}^{m} (\ln (1 + \frac{1}{n})-\frac{1}{n+1})= 1 – \gamma$

I am trying to understand why the Euler-Mascheroni constant $\displaystyle \gamma = \lim_{n \rightarrow \infty} \left ( \sum_{k=1}^n \frac{1}{k} – \ln n \right )$ is equal to $1 – \displaystyle \int_{1}^{\infty} \frac{t-[t]}{t^2}\; \mathrm{d}t$ where $[t]$ is the floor function. There has been an answered question here already: Integral form for the euler-mascheroni gamma constant using […]