Articles of integration

When Are We Allowed to Break Up A Triple Integral?

I was looking over the triple integral below: And I was wondering, when exactly are we allowed to break up a triple integral into the product of its components?

What is the relation between Fourier's Inversion theorem and the Dirac-Delta function?

This is a direct quote from page 472 of this book: From Fourier’s Inversion theorem $$f(t)= \int_{-\infty}^\infty f(u) \, \mathrm{d}{u} \left( \frac{1}{2\pi}\int_{-\infty}^\infty e^{-i\omega(t-{u})} \,\mathrm{d}\omega \right) \tag{1}$$ comparison of $(1)$ with the Dirac-Delta property: $$f(a)= \int f(x) \, \mathrm{d}x \, \delta(x-a)$$ shows we may write the $\delta$ function as $$\delta(t-u)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{i\omega(t-{u})} \, \mathrm{d}\omega$$ My question is […]

f is monotone and the integral is bounded. Prove that $\lim_{x→∞}xf(x)=0$

Question $f : [0,\infty] \to \Bbb R $ is monotone and $\displaystyle∫^∞_0f(x)\,dx$ converges. Note: we also proved before $\lim_{x→∞}f(x)=0$ Show that even $\lim_{x→∞}xf(x)=0$ Thanks!

Definite integral $\int_1^2 \sqrt{1+\left(-x^{-2}+x^2/4\right)^2}\,dx$

I’m having trouble solving this integral. It relates to an arc length question. I tried Wolfram|Alpha, but when it solves it doesn’t give me the option to view step-by-step. Integral: $$ \int_{1}^{2} \sqrt{1+\left(-x^{-2}+\frac{x^2}{4}\right)^2}\,dx $$ Original question: $$ y=\frac{1}{x} + \frac{x^3}{12} $$ Find the arc length of the function on [1,2]. I used the equation: $$ […]

Integration by Parts? – Variable Manipulation

$$\int x^3f”(x^2)\,\mathrm{d}x$$ Solve using Integration by Parts. \begin{align} u&=x^3\qquad\mathrm{d}v=f”(x^2) \\ \mathrm{d}u&=3x^2\qquad v=f'(x^2) \\ &=x^3f'(x)-\int f'(x^2)3x^2 \\ u&=3x^2\quad\mathrm{d}v=f'(x^2) \\ \mathrm{d}u&=6x \qquad v=f(x^2) \\ &=x^3f'(x^2)-[3x^2f(x^2)-\int f(x^2)6x] \end{align} No clue what do from here as the correct answer is: $$\frac{1}{2}(x^2f'(x^2)-f(x^2))+C$$ Can you guys think of anything? I appreciate the help. 🙂

Simple Monte Carlo Integration

I am trying to use Monte Carlo Integration, which is nicely described in the answer here (Confusion about Monte Carlo integration). I am using Monte Carlo Integration to evaluate $\int_0^1x^2\,dx$. I set $w(x) = f(x)/g(x) = x^2/2x = x/2$ Then, I solved for $(1/n)\sum_i^nw(x_n) = (1/n)\sum_i^nx/2$ However, I do not know if this is a […]

Is the integral over a component of a doubly continuous function continuous?

Given a continuous function $$f:(0,1)\times(0,1)\to\Bbb R$$ so that $t\mapsto f(t,x)$ is integrable for all $x$, does it follow that $$x\mapsto\int_0^1 f(t,x)\,dt$$ is continuous in $x$? I would think not necessarily, in part because we are considering open intervals which blocks trivial proofs using uniform continuity. But I cannot think of a counter example.

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 http://www.wolframalpha.com/input/?i=integral++%28%28x^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: $$\displaystyle\int \bigg(\small\sqrt{\normalsize x +\small\sqrt{\normalsize x +\small\sqrt{\normalsize x +\sqrt{x}}}}\;\normalsize\bigg) \;dx$$