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Questions tagged [improper-integrals]

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 $\infty$ or $-\infty$, or as both endpoints approach limits.

An improper integral is defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or $\infty$ or $-\infty$, or as both endpoints approach limits.

Specifically, an improper integral is a limit of the form:

$$\lim_{b\to \infty} \int_{a}^{b} f(x) \ dx \,,\ \lim_{a \to -\infty} \int_{a}^{b} f(x) \ dx$$ or of the form $$\lim_{c \to b^{-}} \int_{a}^{c} f(x) \ dx \,,\ \lim_{c \to a^{+}} \int_{c}^{b} f(x) \ dx$$

in which one takes a limit in one or the other (or sometimes both) endpoints.

Often, we can compute values for improper integrals, even when the function cannot be integrated in the conventional sense (as a Riemann integral, for instance), because of a singularity in the function or because one of the bounds of integration is infinite.

7820 questions
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Convergence of $\int_{-\infty}^\infty \frac{dx}{x^3+1}$

$$\int_{-\infty}^\infty \frac{dx}{x^3+1}$$ Online integral calculator says this definite integral is $\pi/3$ but my textbook says it diverges. Integrand is undefined at $x=-1$ and I don't know if we can use Cauchy principal value.
offret
  • 443
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double integration involving infinity

While studying the plume rise Gaussian Model , I came across following Improper integral which I was unable to solve : $$\int_{0}^\infty \int_{-\infty}^{\infty} \mathrm{exp}[-((y^2+z^2)/2)]\,dy\,dz$$ The source I am referring says to apply the…
Shashank Dwivedi
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Finiteness of Improper integral

How do I show that if $t> 1/2$, then $$\int_R \dfrac{1}{(1+x^2)^t}dx $$ is finite? I tried to use the p-test, but since $\int_R \dfrac{1}{x^p}$ diverges for all $p$, I could not show the finiteness of above integral
nan
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How can I solve this?

$$ \int_{-\infty}^{\infty} \frac{x^2e^{-\alpha x^2}}{x^2+b^2} dx $$
Dinesh Shankar
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determine whether an improper integral converges

Let $b\in\mathbb{R}$, prove or disprove that the improper integral $$ \int_0^\infty x^b \cos(e^x)dx$$ converges. I used Wolfram Alpha to compute several values of $b$, it seems that for $b\geq 0$ this integral converges and $b<0$ it does not. But I…
Chen Ke
  • 489
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Does the Euler spiral - $ \int_{-\infty}^\infty e^{ikx^2} \, dx$ - Converge?

I have a question about this integral: $$ \int_{-\infty}^\infty e^{ikx^2} \, dx = \sqrt{\frac{\pi}{8}}(1+i) $$ Essentially we are following this curve with -- the Cornu spiral: $x = \cos t^2$ $y = \sin t^2$ The Wikipedia article has an image, but…
cactus314
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Show that $\lim_{\sigma \to 0} \int{f(x+\varepsilon)\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{\varepsilon^2}{2\sigma^2}}\mathrm{d}\epsilon} = f(x)$

I would like to show that $$\lim_{\sigma \to 0} \int{f(x+\varepsilon)\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{\varepsilon^2}{2\sigma^2}}\mathrm{d}\varepsilon} = f(x)$$ This is reasonable because…
Sam
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Comparison test with improper integral

I have the integral $$\int_2^\infty\frac{3}{\sqrt[3]x(x+2\sqrt x)}dx$$ and have to find out whether it's divergent or convergent using the comparison test. I've been trying to understand this topic but when I have more than 2 x's in the denominator…
user349556
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Improper convergence of $ \cos(x)/{x^{1/2}} $

I have to evaluate the convergence of the improper integral $ \int_1^\infty \frac {\cos(x)}{x^{1/2}}dx $. As the function is continuous on every $ [1, M] $, I can tell that this function is Riemann integrable on every interval $ [1,M] $, M > 1. So…
aga7689
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For what value does the following improper integral exist

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.
Due_Date
  • 94
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Evaluating $\int_0^{\infty} \frac{\sin(xt)(1-\cos(at))}{t^2} dt$

The problem is to evaluate the improper integral: $I = \int_0^{\infty} \frac{\sin(xt)(1-\cos(at))}{t^2} dt$. This can be written as follows: $$I = \int_0^{\infty} dt \frac{\sin(xt)}t \int_0^a \sin(yt)dy = \int_0^{\infty} dt \int_0^a…
larryh
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How to show the convergence of the integral $\int_{0}^{1}\dfrac{\left(t-1\right)}{\ln t}t^x\mathrm{d}t$?

The integral is defined, for all $x\in\mathbb{R}$ as follows: $$I= \int_{0}^{1}\dfrac{\left(t-1\right)}{\ln t} t^x\mathrm{d}t.$$ When $I$ converges? Let $t-1=u$, we have: $u\to 0$ when $t\to 1$. $$\dfrac{\left(t-1\right)}{\ln…
1-approximation
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Convergence of an integral $\int_1^{+\infty} \frac{1}{x\sqrt[3]{x^2+1}}\mathrm dx$

Convergence of an integral $\int_1^{+\infty} \frac{1}{x\sqrt[3]{x^2+1}}\mathrm dx$ $\int_1^{+\infty} \frac{1}{x\sqrt[3]{x^2+1}}\mathrm dx=\lim\limits_{t\to\infty}\int_1^{t} \frac{1}{x\sqrt[3]{x^2+1}}\mathrm dx$ Partial integration can't solve the…
user300045
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Integral involving an exponential and logarithmic function

I am trying to find the expected value of a probability density function. Solving the integral of the function times its random variable with integration by parts, I arrive at the following integrals which are rather complex. I'd appreciate very…
user283870
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