# Tagged Questions

Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

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### For what values of K, is the integral improper?

For what values of $K$ ($K > 0$), is the following integral improper? $$\int_{0}^{K}\frac{x}{x^2-2}$$ Now, I know that the function is undefined at $x=\sqrt{2}$. I also figured out that the ...
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### How to find the inverse Fourier transfmation of $\exp(-sk)/k$.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of $\exp(-sk)$ which is $$\frac{\sqrt2}{\sqrt \pi}\frac{x}{x^2+ s^2}$$ ...
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### Convergence of $\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$

The problem I'm facing is as it follow: For which values of $a$ the integral converges: $$\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$$ So far I figured that if $a< 1$, the ...
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### Evaluate for $t\in \mathbb{R}$ $\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx$

Evaluate for $t\in \mathbb{R}$ $$\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx.$$ Here is what I have done: Let $f(z)={e^{itz}\over (1+z^2)^2}$. This has two poles $z=i$ $z=-i$ and an essential ...
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### Contour Integration of $\sin^2(x)/(1+x^2)$

How should I calculate this integral $$\int\limits_{-\infty}^\infty\frac{\sin^2x}{(1+x^2)}\,dx\quad?$$ I have tried forming an indented semicircle in the upper half complex plane using the residue ...
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### What's about $\sum_{n=1}^\infty e^{-p_n u}$, where $p_n$ is the nth-prime number?

I am assuming that the following function, for which I am asking as reference request, should be known in the literature, since Glaisher studied the Prime Zeta Function, and my computation is the ...
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### Improper integral with module

faced with a problem when calculating the value of the integral $$\int_{0}^{\infty} e^{-x}|\sin(x)|\, \mathrm{d}x$$ Is there any idea how?
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### Test improper integral with $\ln$ for convergence [closed]

Can you help me to test this integral for convergence, please $$\int\limits_1^e \frac{1}{\sqrt{1 - \ln^2x}}\,dx$$
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### Confusion when finding convergences using divergence and integral test?

I am having a bit of confusion doing the divergence and integral tests, specifically when I am trying to visualize the functions to get a better idea of why the methods work. For example, take the two ...
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### integrating f(x)=1/x from -a to a. convergent or divergent?

we are discussing improper integrals in Calc II, and I am failing to understand why the integral from $-a$ to $a$ of $f(x)=1/x$ is not zero. Since the function is odd and thus symmetric about the ...
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### test for conditional and absolute convergence

find value of a parameter $\alpha$ at which integral converges absolutely and at which conditional $$\int\limits_0^\infty \frac{x + 1}{x ^ {\alpha}}\sin(x)\,dx$$ We can consider 2 cases: area of $0$ ...
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### A sufficient condition for the existence of an improper integral (or a counterexample for it)

Let me try to explain the spirit of the question. The functions $f(x)=1/x^{p}$ for $0<p<1$ and $f(x)=\ln x$ have the following properties: they are in some respect 'nice' on $\mathbb{R}^{+}$, ...
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### What is $\displaystyle\int_{2}^{2}\frac{dx}{x-2}$?

Evaluate the integral: $$\displaystyle\int_{2}^{2}\frac{dx}{x-2}.$$ 1)When does $\displaystyle\int_a^a f(x)dx=0$? Always? 2)Does $\displaystyle\int_a^a$ means area between $(a,a)=\emptyset$? 3) Do ...
Evaluate the integral $$\int\int_{R}(x-y)^2 cos^2(x+y)dxdy$$ where $R$ is the rhombus with successive vertices as $(\pi,0), (2\pi,\pi), (\pi,2\pi), (0,\pi).$ My attempt- I tried doing this surface ...