# 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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### Sequence converges but Improper Integral diverges [on hold]

Can someone give me an example of a sequence such that $$a_{n}=\int_{1}^{n}f(x)dx$$ converges but $$\int_{1}^{∞}f(x)dx$$ diverges.
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### Fourier coefficient and computing an improper integral

I am having difficulties with this problem. I don't really know where to start, I suspect there is something I am supposed to know or "see" that I am missing. $u$ is a $2\pi$-periodic function ...
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### When is the incomplete Beta funtion finite?

I am interested in the incomplete Beta function as defined on Wolfram Mathworld, i.e. $$\text{Beta}(z;a,b)=\int_0^z u^{a-1}(1-u)^{b-1}.$$ I can't seem to find any results on the convergence of this ...
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### Proof of the convergence of $\int_0^\infty\sin{(x^4)} dx$ with Riemann-Lebesgue lemma

In this question, a comment from Lucian asserts that the convergence of the integral $$I=\int_0^\infty\sin{(x^4)} dx$$ is due to the Riemann-Lebesgue lemma. However, I don't immediately see how to ...
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### Improper integral - check convergence

Does the following improper integral converge ? $$\int _0^{\frac{1}{2}}\:\cfrac{1}{\sin\left(x\right)\ln\left(x\right)}dx$$ I have tried to compare it to some known improper integrals but with no ...
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### What is the result of $\int_{0}^{\infty} \frac{1}{x(x+1)}\ln(x+1)dx$

Is there a result for $$\int_{0}^{\infty} \frac{1}{x(x+1)}\ln(x+1)dx$$ If not, is there any upper bound for that? Update: The result is $\frac{\pi^2}{6}$. But how to prove it?
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### improper integral - claim of convergences

Claim: Let $f:[a,b) \to R$ be a non-negative and continuous function. Suppose $\int_{a}^{b} e \ ^{f(t)} dt$ converge , than $\int_{a}^{b} f(t) \ ^ 7dt$ converge too. I think this claim is true. ...
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### Maxwellian integral : is there a closed form?

$f_A(x,y)=\int_0^\infty du \frac{u \left(e^{-\frac{(u-x)^2}{2 A}}-e^{-\frac{(u+x)^2}{2 A}} \right)}{\sqrt{2 \pi } \sqrt{A} x \left(y^2+u^2\right)}$ is there a closed form? I was able to find ...
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### Integral of Product of modified Bessel function, exponential functions and power function

I am trying to evaluate the definite integral 1 to obtain a closed-form solution, with z being the integration variable, and the other parameters are real positive constants. The integral can be ...
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I am trying to compute this integral: $$\int_{0}^{\infty}\prod_{k = 1}^{d}\left(1 - \,\mathrm{e}^{-a_{k}\,t}\right) \,\mathrm{e}^{-t}\,\mathrm{d}t,\quad \mbox{where}\quad a_{k} > 0, \forall\ k.$$ ...
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### Can you prove the convergence of $\int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx$?

Can you prove the following improper integral is convergent? $$\int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx.$$
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### $p$-power integral and $p$-series in higher dimensions

This seems like a basic question that should be addressed in a multivariable calculus course however I don't think I've ever confronted the issue until I became confused about a recent question here ...
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### Finding a specific improper integral on a solution path to a 2 dimensional system of ODEs

In my study of dynamical systems I was recently met with this system of ODEs: $\dot{x}=\frac{\sinh{(y)}}{\cosh{(y)}+A\cos{(x)}}$ $\dot{y}=\frac{A\sin{(x)}}{\cosh{(y)}+A\cos{(x)}}$ for a ...
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### Solving an improper integral contour integral, calculated via Wolfram but in need of analytic derivation possibly

In my studies of dynamical systems I have just encountered this supposedly tough looking improper integral, which is (not really relevant for my predicament) the Melnikov function, with the integral ...
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### Calculus Improper Integral Convergence; Which is right: Limits or Areas?

Could someone please explain to me the following doubt I have on improper integral: $$\int_{-\infty}^{\infty} \frac{1}{x} \ \mathrm{ dx}$$ I still think that since integrals signify areas that this ...
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### how to solve $\int \frac{ e^{4cy^3+2by^2 + (a-3c)y - b}} {\sqrt{1-y^2}} dy$?

here $a,b,c$ are constants it can be solved as indefinite integral or a definite integral with limits [-1,1] or [0,1] MATLAB is not helping here
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### Method for calculating integral of $e^{-2ix\pi\psi}/(1+x^2)$
I am seeking the method for calculating the following integral $$\int_{-\infty}^\infty\frac{e^{-2ix\pi\psi}}{1+x^2} dx$$ Ideas I have are: 1) substition (however which one?) 2) integration by ...