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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.

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Improper integrals Question

I was asked to define the next intergrals and I want to know if I did it right: $$1) \int^\infty_a f(x)dx = \lim_{b \to \infty}\int^b_af(x)dx$$ $$2) \int^b_{-\infty} f(x)dx = \lim_{a \to -\infty}\int^b_af(x)dx$$ $$3) \int^\infty_{-\infty} f(x)dx =…
Ofir Attia
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Can someone please check my answers for these questions regarding improper integrals?

a) Evaluate $$\int^{1}_{0.5} \frac{1}{x^2}dx$$ my answer: $-1+\frac{1}{0.5}=1$ b) Explain why $$\int^{-1}_{1} \frac{1}{x^2}dx$$ is undefined. my answer: since the function $\frac{1}{x^2}$ is undefined at $x=0$, the integral is undefined. c)…
user130306
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How can I evaluate this given improper integral?

How can I evaluate this integral: $$\int _{ 0 }^{3}{ \frac { x }{ (3-x)^{\frac{1}{3}}} dx} \ ?$$
Lia Denisse
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Improper integral $e^{-x-y}$

I am stuck with this problem: $$\int \int_Q e^{-x-y}dA$$ where Q is the first quadrant of the XY plane. I then rewrite it as $\int_0^ \infty \int_0 ^\infty e^{-x-y}dxdy$. So far so good. If we start with the inner integral we get $\int_0 ^\infty…
J. Doe
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Convergence of $\int^{\pi/2}_0 x\sqrt{\sec x}dx$

At $x=\pi/2$, $\sec x$ goes to infinity, and $x$ is fixed, so $x\sqrt{\sec x}$ goes to infinity. It seems to diverge, but the solution says it converges. I don't know how to prove it. I cannot find the antiderivative of this function or suitable…
user533661
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Inequality involving an improper integral

To prove: $\displaystyle \int_x^\infty \exp \left(-\frac{t^2}{2}\right) \, dt < \frac{1}{x}\exp\left(-\frac{x^2}{2}\right)$, $\quad x>0$ We know $$\int_x^\infty t^{-2} \exp\left(-\frac{t^2}{2}\right) \, dt \leq…
Vinay Deshpande
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Convergence of improper integral $\int_{0}^{+\infty}\frac{\arctan \alpha x - \arctan \beta x}{x} dx$

I have to analyse the convergence of $\displaystyle \int_{0}^{+\infty}\frac{\arctan \alpha x - \arctan \beta x}{x} dx$; $\alpha,\beta \in R$ I have written: $\displaystyle \int_{0}^{+\infty}\frac{\arctan \alpha x - \arctan \beta x}{x} dx =…
zdikov
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Convergence of $\int_1^\infty(\cos^2(\pi x))^x\ dx$

I am looking for ways to figure out whether the integral $$\int_1^\infty(\cos^2(\pi x))^x\ dx$$ converges. If not, are there other similar integrands, for example $$\int_1^\infty(\cos^2(\pi x))^{x^x}\ dx$$ for which the integral does converge?
Alvin L-B
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A majoration (integral inequality)

How can we show that for fixed $a>0$ and $\forall\zeta \in (0,a)$ it holds $$\int_0^\infty e^{-at^{2}}t^{2n}dt\leq \frac{1}{2}\frac{n!}{\zeta^{n}}\sqrt{\frac{\pi}{a-\zeta}}$$ where $n$ is a non negative integer? Any hint or help is appreciated to…
User6797
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How to test this improper integral is convergent?

The question is to test whether $$\int^\infty_1\frac{x^{2012}-20x^7-14}{x^{2014}-30x^{11}+13}dx$$convergent or not. I have found that this rational function which is integrated from 2 to infinity is convergent, and I also founded that there is only…
spencer Dai
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Show $\lim_{x\to +\infty} \int_1^x \frac 1{x^\alpha+t^\beta}\;\mathrm{d} t = 0$.

Show that $$ \lim_{x\to +\infty} \int_1^x \frac 1{x^\alpha+t^\beta}\;\mathrm{d} t = 0 $$ if $\alpha, \beta >0$ and $\max{\alpha, \beta} > 1$.
Wei Zhong
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How is this improper integral solved?

I'm reading through the lecture notes of Wayne Hu regarding the Damping Scale of the CMB. He give the following steps to calculating the damping scale, $k_D$:$$k_D^{-2}=\int \frac{1}{6(1+R)}\left( \frac {16}{15}+…
Quark Soup
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Rate of decreasing sleeping time question

can someone help me solve this problem? It is estimated that $t$-weeks into a semester, the average amount of sleep a college math student gets per day $S(t)$ decreases at a rate of $$\frac{-6t}{e^{t^2}}$$ hours per day. When the semester begins,…
Hami the Penguin
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Help evaluating convergence of an integral with parameter $a$

So I have this improper integral: $$\int_a^{\infty}\frac{1}{x(x^2-1)^{6a}}dx$$ So I need to evaluate(find intervals for) convergence regarding parameter $a$. So i know the following convergence rule: $$\int_{{\,a}}^{{\,\infty…
MathIsTheWayOfLife
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$\int_0^\infty |f|^2<\infty, \int_0^\infty |f''|^2<\infty$, then $\int_0^\infty |f'|^2<\infty$.

If $f$ is twice continuously differentiable on $[0,+\infty)$, and $\int_0^\infty |f|^2<\infty, \int_0^\infty |f''|^2<\infty$, then $ \int_0^\infty |f'|^2<\infty$? My attempt: $\int |f'|^2 =\int f'd f =ff'|_0^\infty-\int_0^\infty f'' f $. The last…
xldd
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