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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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Show $ \int_0^\infty \frac{\sin ax \sin bx}{x^2} \, dx = \frac{\pi a}{2} $ if $ a < b$

Show that if $a < b$: $$ \int_0^\infty \frac{\sin ax \sin bx}{x^2} \, dx = \frac{\pi a}{2} $$ I could see solution by Fourier analysis or by Contour Integration, but why does the larger frequency $b$ not matter?
cactus314
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improper integrals( very basic)

I don't quite understand what it really means. Do it mean that $f(c_i)$ could be infinite and $dx$ is very small, so you can't determine what the infinite$\times$very small is?
whoisit
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Convergence of improper integral $\int_{2}^{\infty} \frac{1}{log(t)}dt$

Convergence of improper integral $\int_{2}^{\infty} \frac{1}{log(t)}dt$ How do i start?
Taylor Ted
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To test convergence of improper integral $ \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\, \mathrm dx$

I have to test convergence of improper integral $$ \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\,\mathrm dx$$ I write as $\log(x) \leq x$ . So $x\log(x) \leq x^2$. So $ \frac{x\log(x)}{(1+x^2)^2} \leq \frac{x^2}{(1+x^2)^2}$ . Now using comparison…
Taylor Ted
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Improper integrals - where am I going wrong?

The question asks: $\int_0^\pi \frac{sinx}{\sqrt{|cosx|}}$ My attempt: $\lim\limits_{t \to \frac{\pi}{2}^-} \int_0^t \frac{sinx}{\sqrt{|cosx|}}$ + $\lim\limits_{t \to \frac{\pi}{2}^+} \int_t^\pi \frac{sinx}{\sqrt{|cosx|}}$ $\int…
abruzzi26
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Integration by parts with poles at the boundary $\int_0^\infty f(x)/x \; d x$

Let $$f : (-\infty, \infty) \rightarrow \mathbb{C}$$ be an even ,smooth, and compactly supported, vanishing at zero. Is there a way to solve integral $\int_0^\infty f'(x)/x \; d x$ in a suitable interpretation a la integration by parts.
Marc Palm
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How to evaluate an improper integral.

I don't conceive how to evaluate the improper integral: $$ \int_{1}^{\infty}\frac{\sin x}{\sqrt{x-1}}dx. $$ When this converges, I'm glad if you give the value of this integral. Thank you.
user
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Calculus $\int_0^{+\infty}\frac{\sin^2x}x\mathrm dx$

Calculus $$\int_0^{+\infty}\frac{\sin^2x}x\mathrm dx$$ I have just approached to improper integrals, and it may be rather complex to me.
mja
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limit of an improper integral by comparison theorem

I am studying an integral using comparison text. I have managed to show it easily that for $b < 1$, $$\int_0^1 \frac{ln(1+x)}{x^b}dx$$ convergens and this is because I am well aware of functions that behave a certain way when $b$ is either less…
Mikale Larson
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Interchange of limit operator and $\ln$ function.

$$\lim_{n\to \infty}\ln\left(\frac{1+a^2n^2}{1+n^2}\right)$$ Can someone help evaluate that for me?
user2250537
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Improper Integral of $xe^{-x}$.

I was working on this problem but I didn't get the right answer, though I can't find my mistake. Here is the question and my attempt: $\int_a^\infty xe^{-x}dx$ evaluate. $\lim_{b\to \infty} \int_a^b xe^{-x}dx$ then using parts, by letting $u = x$…
user2250537
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Improper double integral

Can I apply the Fundamental Theorem of Calculus for $$\int_{-\infty}^{t_1} \int_{-\infty}^{t_2} \frac{\partial \phi\left(\frac{z_2 - \rho z_1}{\sqrt{1 - \rho^2}}\right)}{\partial z_2} dz_2 dz_1$$ in order to get $$\int_{-\infty}^{t_1}…
Kolibris
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How do you evaluate an exponential term that contains both $-\infty$ and $+\infty$?

What does $\int_{0}^{\infty} e^{y(iu-\alpha)}dy = ?$ Please note $i$ is a complex variable, $\alpha$ and $u $ are constants. I know this integral evaluates to: $$\left.\frac{e^{y(iu-\alpha)}}{iu-\alpha}\right|_0^\infty$$ I'm unsure how this…
user11460
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Value of convergence of $\displaystyle\int\frac{\sqrt{x}}{1+x^2}$

How to prove that converge $$\int^{\infty}_1\frac{\sqrt{x}}{1+x^2}$$ and find this value.
Roiner Segura Cubero
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Evaluating Improper integral

Using the equation $\frac{1}{\sqrt{x}}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}\exp(-\alpha^{2}x) \, \mathrm{d}\alpha$, for $\alpha>0$, Compute the two integrals $$\int_{0}^{\infty} \frac{\cos x}{\sqrt{x}} \, \mathrm{d}x \quad \text{and} \quad…
user107723
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