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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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 ...
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Surface Integral over a rhombus

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 ...
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Find the value of p for which the integral converges and evaluate integral for $\int ^\infty_e \frac{1}{x(\ln x)^p} dx$

Find the value of p for which the integral converges and evaluate integral for $\int ^\infty_e \frac{1}{x(\ln x)^p} dx$ MY ATTEMPT: Given Integral: $I= \int ^\infty_e \frac{1}{x(\ln x)^p} dx$ put ...
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Integral of reciprocal of absolute value

I am having trouble with the following question. For which values of $n, p$ is the integral $\int_{\mathbb{R}^n}\frac{1}{|x|^p}$ convergent? I need to derive a relationship between $n$ and $p$, i.e. ...
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Can $\int_0^1\frac{1}{t}e^{-t} dt$ be analytically or numerically integrated?

The following integral has a singularity at $t = 0$ as in this situation the exponential term becomes $1$ and it no longer dominates the $\frac{1}{t}$ term. $$f(x) = \int_0^1\frac{1}{t}e^{-t}dt$$ So ...
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Divergence of an integral

I am sure this is a familiar example to many, as it comes up as an example of a function which belongs to $L^p$ for $p=1$ but not $L^p$ for $p < 1$, but I am having a hard time seeing why it is "...
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Name of a particular improper integral

I am curious if there is a particular name for this, $\int\limits_{-\infty}^\infty e^{i\xi^2}d\xi$. I think it might be related the Fresnel integral but I cannot see it, any suggestions?
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Seemingly Obvious Improper Integral Property

The following seems extremely obvious, so much so that I cannot see how to prove it: If $f:[a,b]\to\mathbb{R}$ is Riemann integrable then $$\int_a^bf(x)dx=\lim_{c\to b^-}\int_a^cf(x)dx$$
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Evaluate $\displaystyle\int\limits_0^{\infty}\frac x{20}e^{-x/20} dx$

I tried doing $u$-substitution and got $-20e$ as my final answer, but I think the correct answer is just $20$. I'm not sure what I did wrong, but probably had to do with plugging in infinity... could ...
Show the integral is convergent and find the value it converges to. $$\int_1^\infty \frac{\arctan x}{x^2}$$ I have found the indefinite integral to be -\frac{\arctan x}{x} + \ln|x| -\frac{1}{2}\... 0answers 48 views Why the integral converges? Say we take one of the loveliest functions in mathematics the Gaussian which looks like this: Picture of Gaussian. By eye inspection we can say that this looks like something that could have finite ... 0answers 34 views Why do the integral and the partial sum agree for small a and m? Consider the following naive manipulations: \begin{align} \int_0^\infty \frac{e^{-x}}{1+ax}\:dx & = \int_0^\infty e^{-x}\frac{1}{1-(-ax)}\:dx\\ &= \int_0^\infty e^{-x} \left( \sum_{n=0}^\... 2answers 32 views Improper Integral of (y-1)^{-3/2} from 0 to 2 improper integral\int_{0}^{2} \frac{1}{(y-1)^{3/2}}\, dy$$I know it doesn't work when y = 1, so I split the integral, right. But then i realized, it doesn't work with 0 either, as that would ... 1answer 28 views Laplace transform identity F(s) = \mathcal{L}(t^{-3/2} \mathrm{e}^{-1/t}) I'm asked to prove the following result: If F(s) is the Laplace transform of f(t) = t^{-3/2} \mathrm{e}^{-1/t}, show that F'(s)=-s^{-1/2}F(s). I'm having a lot of troubles to prove this ... 0answers 24 views Discussion on convergence of improper integrals I have a general question about the convergence of improper integrals and it is this: If we have an improper integral that converges, is this similar or analogous at all to saying that a function is ... 1answer 40 views Improper convergence of this integral?$$ \int_1^{\infty} \left\langle t\right\rangle\dfrac{\cos\left(t\right) - \sin\left(t\right)}{t^2}\,dt  where $\left\langle t\right\rangle$ is the rationale part of $t$. I would like to use the ...
Find $m$ such that $\displaystyle\int_{-\infty}^\infty \frac{1}{(1+x^2)^m} \, dx$ is finite. I tried to substitute $x$ with $\tan\theta$ but got stuck.