# Tagged Questions

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### Improper integral $\int_{B}\frac {1}{|x|^\alpha}dV$

Let B be the ball $|x|\le 1$, $x\in R^n$. For what $\alpha$ does $$\int_{B}\frac {1}{|x|^\alpha}dV$$ exists? I find it hard when it comes to generalize this statement in $R^n$. I've been able to do ...
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### Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
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### Improper double integral evaluation by changing the order of integration

I was watching an MIT OCW recitation video about exchanging the order of integration on double integrals. The example was: ...
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### Show $\iint xye^{-xy}\,dx\,dy$ is convergent or divergent

Determine convergence/divergence of $$\iint xye^{-xy}\,dx\,dy$$ for $x,y \geqslant 0$ i.e. in the first quadrant. I have managed to show that $xye^{-xy} \to 0$ in the first quadrant but other ...
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### Integration limits of the double integral after conversion to the polar coordinates

I want to solve the following double integral: $$\int_0^{\infty}dx\int_{-\infty}^{\infty}dy\,f(x,y).$$ And for example I made a conversion to the polar coordinates, $x=r\cos{\theta}$ and ...
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### Evaluation of a particular type of integral involving logs and trigonometric function

Is there any closed form for $$\int _0 ^{\infty}\int _0 ^{\infty}\int _0 ^{\infty} \log(x)\log(y)\log(z)\cos(x^2+y^2+z^2)dzdydx$$ if yes then how to prove it?
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### Is $\iint\limits_{D}\frac{1}{(x^{2}+y^{2})^{5}+5}dx\,dy$ convergent in $D=[0,+\infty)\times[0,+\infty)$

Can you tell me whether my approach is correct. We first switch to polar coordinates and we get the following integral: ...
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### Improper Multiple Integral

I am trying to solve the following exam problem: Let $s$ be a real number. Find the condition under which the improper integral $$I:=\iint_{\mathbb R^2} \frac{dxdy}{(x^2-xy+y + 1)^s}$$ converges, ...
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### how to find this type of definite double integral?

could any one tell me how to find this type of definite double integral? $$\int_{0}^{\infty}\int_{x}^{\infty}{e^{{-y\over2}}\over y}dydx$$ Thank you.
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### When are the following multiple improper integrals convergent?

This is a question from a past exam. For which $p, q\in \mathbb R$ do the following improper integrals converge? ...
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### A parameterized elliptical integral (Legendre Elliptical Integral)

$$K(a,\theta)=\int_{0}^{\infty}\frac{t^{-a}}{1+2t\cos(\theta)+t^{2}}dt$$ For $$-1<a<1;$$ $$-\pi<\theta<\pi$$ I know this integral to be a known tabulated Legendre elliptic integral, ...
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### Converging/Diverging Integrals

Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent": $$\int_4^\infty \frac{1}{x^2+1}\,dx.$$ I am so lost. I ...
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### Question about L2 Inner Product and Integrals

Does exists $f\in L_2(\mathbb{R}^d)$ such that for all $g\in L_2(\mathbb{R}^d)$ which is not identically zero: ...
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### Change of variables in improper double integral - how to find the limits?

Consider the following integral (originating from the product of the Laplace transforms of $f$ and $g$): $$\int_0^\infty \int_0^\infty f(u)\ g(v) e^{-s(u+v)}\ du\ dv.$$ For this integral, the ...
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### Determine $\alpha >0$ for which $\iint_Af(x,y)^\alpha dx \, dy < +\infty$

Let $A=\{(x,y)\in \mathbb R^2: 0<x<1, 0< y < \sqrt{x}\}$ and $f \colon A \to\mathbb R$ a continuous function s.t. $$\frac{1}{x^2+y^2} \le f(x,y) \le\frac{2}{x^2+y^2}$$ for every ...
In Exercise 5 (f) of Angus Taylor's Advanced calculus (p. 659) one is asked to find the value of the following integral if convergent: I:=\underset{R}{\iiint}\dfrac{x^2 y^2 z^2}{r^{17/2}}\mathrm ...