# 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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### Integrals of a function and its absolute value

Is the following proposition true? Let $f(x)$ be a real-valued function defined on $[a,b] \subset \mathbb{R}$, and suppose that the integral, $$I = \int_a^b f(x) dx,$$ exists in the sense of ...
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### Juantheron-like integral

While seeing this post, the following integral is just struck me $$\int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1$$ I have tried like what user @OlivierOloa did in ...
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### Multi integral change of variables?

Sketch the domain D bounded by $y=x^2$, $y=1/2x^2$, and $y=2x$. Use a change of variables with the map $x=uv$, $y=u^2$ to calculate: $∫∫y^{-1}dxdy$ Ok so I found the Jacobian to be $-2u$, and ...
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### Using ML inequality

I'm trying to work out the following integral: $$\int_{-\infty}^\infty \frac{1}{\cosh(x)} dx$$ by complex analysis methods and as a result of the contour I chose (a rectangle) one of the things I ...
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### Understanding an apparent contradiction from naively applying Taylor's theorem and Fubini's theorem together

Suppose $f(x)$ is a bounded, $\mathcal{C}^{\infty}$ function on $\mathbb{R}$ for which the integral $$I = \int_{0}^{\infty}f(x)\ dx,$$ exists. Taylor's theorem implies $f$ admits a MacLaurin expansion ...
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### On series of the kind $\sum_{n=1}^\infty\frac{1}{n^2}\cdot\frac{1}{(1+nx)^s}$ and Frullani's theorem

I would like to know if my computations, in this post are not required justifications for the computations unless if there is a mistake in my claims, were rights and solve a question as if are known a ...
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### Improper integral $\int_{-\infty}^{\infty} \frac{e^{sx}}{1+e^{x}} dx$

I'm having trouble solving this problem and was hoping someone could show me how it's done? $$\int_{-\infty}^{\infty} \frac{e^{sx}}{1+e^{x}} dx$$ where $0<Re(s)<1$
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### Doubts concerning to an application of Frullani's theorem to $f_k(x)=\frac{2^{-k^2x}-2^{-(k^2+1)x}}{x}$, and Lebesgue convergence theorems

By application of Frullani's theorem for $a_n=n^2+1$, $b_n=n^2$ where $n\geq 2$ and $f(x)=2^{-x}$ then RHS in Frullani's integral is obtained for $n\geq 2$ as $$\log(1-\frac{1}{n^2}),$$ thus I asked ...
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### Integrating lower incomplete gamma function raised to the power $k$

I'm trying to solve the following integral: $$\int_0^\infty \gamma(t,x)^k x^t e^{-x} \mathrm{d} x$$ I'm fighting with it for quiet a while and didn't get any result. Though, I do have the ...
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### Is $\frac{1}{x^2}$ Lebesgue integrable while $\frac{1}{x}$ is not?

My textbook defined integrability as $f$ is said to be Lebesgue integrable if $\int{}f$ is finite. I heard that $\frac1x$ is not Lebesgue integrable, but $\frac{1}{x^2}$ is Lebesgue integrable. I do ...
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### Finding limit with improper integral

How should I approach this question? $$\lim_{x\to0}\frac{1}{x}\int_1^{1+x}\frac{\cos t}{t} \, dt$$ I tried to use L'hospital and that gave me $-\sin(0) = 0$ The correct answer is $\cos 1$. Did I L'...
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### Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
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