Articles of integration

$\int_{1}^{\infty} h(x)\ dx$ converges $\Rightarrow$ $h$ is bounded in $[1, \infty)$

Let $h:[1, \infty)\rightarrow \mathbb R$ a continuous non-negative function, such that $\int_{1}^{\infty} h(x)\ dx$ converges. does $h$ must be bounded in $[1, \infty)$? I tried to prove it by showing that if $\int_{1}^{\infty} h(x)\ dx$ converges, then by the definition the $\lim_{b \to \infty}\int_{1}^{b} h(x)\ dx$ exists. Can I conclude that from the existence of […]

Reducing $\int_0^\infty x^n e^{-ax}\,\mathrm{d}x$

For positive integer $n$, an integral of the form $\int_0^\infty x^n e^{-ax}\,\mathrm{d}x$ will reduce to $\frac{n!}{a^{n+1}}$. This can be shown by successively applying integration by parts and observing that the $\left.f(x)g(x)\right|_0^\infty$ terms vanish. The first such term $f(x)g(x)$ is $\left.-\frac{1}{a}e^{-ax}x^n\right|_0^\infty$. It is clear that an exponential will “beat” any power function as $x$ tends to […]

Setting up integral in polar coordinate for surface area of paraboloid

$$ z = 16 – 3x^2 – 3y^2 $$ above plane z = 4 I tried turning z into $$ z= 16 – 3r^2 $$ but it doesnt seem to be correct

Contour integration of $\frac{\log( x)}{x^2+a^2}$

This question already has an answer here: Evaluate $\int_0^{\infty} \frac{\log(x)dx}{x^2+a^2}$ using contour integration 5 answers

Breaking a contour integral into 3 separate contours?

We can try to integrate the following function around a counter-clockwise circular contour: $$\frac{x^3}{(x-1)(x-2)(x-3)}$$ Can someone show how to use the Cauchy–Goursat theorem (explained here and here) to break this apart into 3 separate contours? In other words, I’d like to take $$\int_c{\frac{x^3}{(x-1)(x-2)(x-3)}}$$ and get $$\int_{C_1}{f_1(x)} + \int_{C_2}{f_2(x)} + \int_{C_3}{f_3(x)}$$ I’m hoping for a pretty […]

Absolute and Conditional Convergence of the integral $\frac{\sin(x)}{x^p}$ for real values of $p$

This question already has an answer here: Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$ 5 answers

Prove or disprove that the given expression is “always” positive

I have previously asked a question and I tried to solve it by my own and it led to the question below: Prove or disprove that $$\small\int_{\mathbb{R}}l(y)^xf_0(y)\mathrm{d}y\int_{\mathbb{R}}f_0(y)l(y)^x\ln (l(y)^x)\ln (l(y))\mathrm{d}y-\int_{\mathbb{R}}l(y)^xf_0(y)\ln (l(y))\mathrm{d}y\int_{\mathbb{R}}f_0(y)l(y)^x\ln (l(y)^x)\mathrm{d}y$$ is greater than $0$. Given: $\rightarrow f_0$ and $f_1$ are some density functions $\rightarrow l(y)=\frac{f_1(y)}{f_0(y)}$ is an increasing function. $\rightarrow x\in(0,1)$

Finding area of a triangle with integrals

I am suppose to find the area of a triangle using integrals with vertices 0,0 1,2 and 3,1 This gives me $y= 2x$ $y=\frac{1}{3}x$ $y= \frac{-1}{2}x+\frac{5}{2}$ for my slopes I know that I can calculate the area of the first part by finding $$\int_{0}^{1}2x-\frac{1}{3}x$$ and the second part by $$\int_{1}^{3}\frac{-1}{2}x+\frac{5}{2}-\frac{1}{3}x$$ The anti derivative of the […]

Residue theorem on even function integration

I need to integrate below function; $$\int_{-\infty}^{\infty} \frac{\sin(pR)}{R}\frac{p}{k^2-p^2} dp$$ here $k,R$ are constants. Since this is an even function of $p$, I tried applying the residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ipR}-e^{-ipR}}{2iR}\frac{p}{k^2-p^2} dp$$ Now taking $z=pR, dz=R dp$; $$\int \frac{e^{iz}-e^{-iz}}{2iR}\frac{z}{k^2R^2-z^2} dz$$ $$\frac{\pi i}{2i}\frac{1}{R}[{(\lim{z \rightarrow kR})} \ \frac{(e^{iz}-e^{-iz})z}{k^2R^2-z^2}(z-kR) +{(\lim{z \rightarrow -kR})} \frac{(e^{iz}-e^{-iz})z}{k^2R^2-z^2}(z+kR)]$$ This gives me zero as the answer. […]

Dyson-expansion like multidimensional integral

Let $n \ge 1$ be an integer. Now let $0 \le t_0 \le t$ and $\beta \neq 1$ be real numbers. Now, let $\vec{p} := (p_0,p_1,\cdots,p_n)$ be strictly positive integers. Also let $(x)_{(n)} := x(x-1)\cdot \dots \cdot (x-n+1)$ be the Pochhammer symbol and let $s(p,k)$ by the Stirling number of the first kind. By generalizing […]