# 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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### 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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### Evaluating real integral by complex contour method

Please let me know where my mistake could be. I've verified the integral $$\int_{-\infty}^\infty \frac{dt}{(t^2+1)(t^2+4)}$$ to be equal to $\frac{\pi}{6}$ with a computer math system. However, I'm ...
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### For which $\alpha$, $\beta$ does $\int\limits_1^{\infty} x^{\alpha} \cdot (\ln x)^\beta dx$ converge? [duplicate]

For which $\alpha$ and $\beta$ does the following integral converge ?: $$\int_{1}^{\infty}x^{\alpha}\,\ln^{\beta}\left(x\right)\,\mathrm{d}x$$ Here is my analysis: I noticed that the function ...
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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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### 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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### 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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### Is this a finite integral?

A book on probability theory I am reading asserts the following: for $x, y \in \mathbb{R}^n, t \geq 0$, consider the function $h_t(x, y) = \frac{1}{t^{n/2}}e^{-\frac{r^2}{4t}}$, where $r = |x - y|$. ...
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### Evaluate the integral $\int_{0}^{+\infty}\frac{\arctan \pi x-\arctan x}{x}dx$

Compute improper integral : $\displaystyle I=\int\limits_{0}^{+\infty}\dfrac{\arctan \pi x-\arctan x}{x}dx$.
### 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 ...