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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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Exponential integral

I have to evaluate $$\int_{-\infty}^\infty dx \int_{-\infty}^\infty dy\quad e^{-k(x^2+y^2+xy/4)}$$. I tried to transform it using $x=(\alpha+\beta)/\sqrt{2}$ and $y=(\alpha-\beta)/\sqrt{2}$. The integrand became $\dfrac{9\alpha^2+7\beta^2}{8}$ but…
Purushothaman
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The improper integral of $\int_{-1}^1 \frac{x-1}{ \sqrt[3]{x^5} } dx$

I have a kind request to check whether my solution is correct. $$ \int \limits_{-1}^1 \frac{x-1}{ \sqrt[3]{x^5} } dx = \int \limits_{-1}^0 \frac{x-1}{ \sqrt[3]{x^5} } dx + \int \limits_{0}^1 \frac{x-1}{ \sqrt[3]{x^5} } dx = $$ $$ =\lim \limits_{A…
Elizabeth_Banks
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How to do this improper integrals??

I'm trying to solve improper integrals but some patterns i can't do anything... This question is patterns of them. please help to to solve this question It's okay that helping just one of them. I really need somebody's help... questions is "State…
daysix
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$\int_{\mathbf{R}} F^n(ax)f(x)\mathrm{d}x$ with $a>0$ and $f=F^\prime$

Fix an integer $n\ge 2$ and let $f: \mathbf{R}\to [0,\infty)$ be a density function, that is, a continuous function such that $\int_{\mathbf{R}}f(x)\mathrm{d}x=1$. In addition, for each $x \in \mathbf{R}$, set $F(x):=\int_{-\infty}^x…
Paolo Leonetti
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A Question on the Solution of a This Improper Integral

I'm studying improper integrals, studying the equation pictured below: When evaluating the third equation, with $\lim_{b\rightarrow\infty} -\frac{1}{u}$ with $\ln b$ as the upper limit and $u = 1$ the lower limit, wouldn't that come out to …
Stanley Yelnates
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An improper integral with Fourier series

I wanna evaluate this integral $$I=\int_{0}^{+\infty}{\exp(\cos(x))\sin(\sin(x))\over x}dx$$ so i wrote $g(x)=\exp(\cos(x))\sin(\sin(x))$ as a Fourier series and I found out that $$g(x)=\sum_{i=0}^{+\infty} {\sin(nx)\over n!}$$ and then dividing…
Mossaab Sama
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Question about convergence of an improper integral

So I have an improper integral: $$\int_{1}^{\infty} \frac{1}{x^a(x-1)^b} dx$$ I have to find for which $a,b\in \mathbb{R}$ it converges. So my initial step is: $$\int_{1}^{\infty} \frac{1}{x^a(x-1)^b} \leq ?\int_{1}^{\infty} \frac{1}{x^ax^b}dx$$ But…
MathIsTheWayOfLife
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Factorization to solve improper integral with exponentials

How calculate $$\int_{0}^{\infty} \dfrac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx$$ with $a$, $b$ are positive values? I think that there is some algebraic manipulation that I can not see.
Luciano Magrini
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A convergent improper integral with $\int k\chi_{\{f>k\}}$ divergent

If $\int_0^\infty f(x) dx\leq\infty$ , we regard it as "convergent" , does there exist such an $f$, which satisfies $\int_0^\infty f(x) dx<\infty$ , but $\int_0^{\infty}k\chi_{E_k}(x)dx$ "divergent" ? Where $E_k=\{x:f(x)>k\}$ Well , I came to this…
Alexander Lau
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Help solving an improper integral

The assignment is to solve: $$ \int_{0}^{\infty}f(x) dx $$ where $$f(x) = \frac{4}{(x+1)^2(x+3)} $$ . I did partial fraction on the indefinite integral of $f(x)$ and got: $$ f(x) = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x+3} $$ $$ 4=(A+C)x^3 +…
user644361
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property about improper integrals

Let be $f:[0,\infty[ \rightarrow \mathbb{R}$ a decreasing and continuos, $f(x)>0 , \forall x$. If $\displaystyle \int_0^{\infty}f(x)dx$ converges then $\displaystyle \lim_{x \rightarrow \infty} xf(x) = 0.$ How to prove that? Ok, I know that if $G(x)…
Joãonani
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How to prove $\int_{0}^{1} \frac{\sin(x)^2}{x^{5/2}}$ is divergent

The original problem is to test the convergence of $\int_{0}^{\infty} \frac{\sin(x)^2}{x^{5/2}}$. It is easy to prove that $\int_{1}^{\infty} \frac{\sin(x)^2}{x^{5/2}}$ is convergent, but I have checked in https://www.integral-calculator.com that…
Andarrkor
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question about improper integrals and limits

Let be $f:[0,\infty[ \rightarrow \mathbb{R}$ continuous. If $\displaystyle \int_0^{\infty} f(x) dx $ converges so $\displaystyle \lim_{r\rightarrow \infty} \int_r^{\infty} f(x) dx=0$ My Attempt: Call $G(r) = \displaystyle \int_0^{r} f(x)dx,…
Joãonani
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State whether the following improper integral converges, and, if it does, find its value.

Hello, I have been struggling with the simplest question of this topic. The integral of it is $\frac{-1}{(1+x)} dx$, and replacing infinity with the limit of $b \to \infty$, I found $\frac{-1}{(1+b)} +1/2$, and the limit goes to zero so the value…
Hami the Penguin
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Does $\int_e^\infty{\frac{\ln(x)^\alpha}{x}dx}$ exists?

My task is to determine wetherthe integral $\int_e^\infty{\frac{\ln(x)^\alpha}{x}\,dx}$ does exist or not in depencence of $\alpha\in \mathbb{R}$. For that I wrote $b$ instead of $\infty$ and then calculated the integral using substitution, which…
Yefexem
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