# Tagged Questions

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### Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$

Compute $$\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$$ The answer is $\pi/2$. The discontinuities at $\pm1$ are removable since the limit exists at those points.
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### Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$

Evaluate $\int_{-\infty}^\infty x\exp(-x^2/2)\sin(\xi x)\ \mathrm dx$ The answer given by Wolfram Alpha is $\sqrt{2\pi}\xi\exp(-\xi^2/2)$. Observe how this is related to the Fourier transform of ...
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### Integral from zero to infinity of $\int_0^{\infty}\frac{(1-e^{-\lambda z})}{\lambda^{a+1}} d \lambda$

I know that the value of the integral is as follows $$\int_0^{\infty}\frac{(1-e^{-\lambda z})}{\lambda^{a+1}} d \lambda =z^a \frac{\Gamma(1-a)}{a}$$ However, how exactly the integral is calculated? ...
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### Calculation of an integral via residue.

$$\int_{-\infty}^{\infty}{{\rm d}x \over 1 + x^{2n}}$$ How to calculate this integral? I guess I need to use residue. But I looked at its solution. But it seems too complicated to me. Thus, I asked ...
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### $\dfrac{\sin x}{x}$ modified improper integrals.

I am trying to evaluate this integrals: $$\int_{-\infty}^{\infty} \! \left[\frac{\sin\left(x\right)}{x}\right]^n \, \mathrm{d}x.$$ I know how to prove it if $n=1$ using Fourier Transform, but I ...
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### How to prove that this integral converges absolutely?

$f:[a,{\infty}[\to\mathbb{R}\$ is bounded and suppose that $f$ is integrable on each interval of the form $[a,b[$. Prove that $$\int_0^\infty \frac { f(x) }{ x^p } \ \, dx$$ converges absolutely ...
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### Problems with determining convergence of integral

It should be easy but I'm not sure... For which $\alpha \in \mathbb{R}$ the following integral is convergent: $$\int_0^1 \int_0^1 \frac{1}{|y-x|^\alpha}dxdy \ \ ?$$ I get for all $\alpha \neq 1,2$ ...
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### Integration of hyperreal functions / Intermediate Value Theorem

Here's a statement on hyperreal function I've been trying to prove (I came up with it but I think it is true): Suppose $f(x)$ is a continuous real-valued function and $h(x)$ is a continuous ...
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### Definite integral involving hyperbolic cosine

I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
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### Computation of a certain integral

I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
My calculus book says that the integral of $\frac1x$ cannot cross zero. Now it seems obvious that because of symmetry, there will always be an interval whose integrals are equal in magnitude and ...
### Assess the limit: $\lim_{n\to\infty} \frac{1}{n}\int_0^n \frac{\arctan(x)}{\arctan{\frac{n}{x^2-nx+1}}}dx$
Compute the following limit: $$\lim_{n\to\infty} \frac{1}{n}\int_0^n \frac{\arctan(x)}{\arctan{\frac{n}{x^2-nx+1}}}dx$$ I'm looking for an easy approach if possible.