# Tagged Questions

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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### Integration indefinite integral of multiple functions

I need help integrating $$\frac{x}{1-\exp(-x^2/a^2)}\exp((x-u)^2/2s^2)$$ wrt $x$, where $a$ and $u$ are constants
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### Is this proof correct? Divergence of $\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \, \mathrm{d}x$

Problem: Show that $$\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \,\mathrm{d}x$$ diverges. I know that there are many questions in which this problem is solved, but I want to know if my ...
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### Help on the Integration of $\int_0^{\infty} e^{-bx}\sin ax^2 \, \mathrm{d}x$.

I have had the misfortune of coming across the following integral, for real $b$ and $a > 0$: $$\int\limits_{0}^{\infty} e^{-bx} \sin\left(ax^{2}\right) \, \mathrm{d}x.\tag{1}$$ Naturally, I ...
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### Does this integral variable change makes sense to you?

I was Reading a book about calculus when I've found this part about variable substitution in integrals: Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its ...
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### Strong Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
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### Integral vs antiderivative

I have a similar question to this one: Integrable or antiderivative. If a function has an antiderivative, does the difference of values of the antiderivative on the endpoints of an interval always ...
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### Hard integration problems book, special functions

I want hard integration problems which level is college competition or harder. I want problems book about hard integration. Would you recommend some problems books? And can you recommend books ...
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### Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
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### Integration over time by having derivation

Assume we want to find the following integration: $$\int_{t=0}^{\infty} p(t)dt$$ where $p(0)=p$ and also $$\frac{dp(t)}{dt}=-p(t)(1-p(t))\mu$$. Is there any easy way to ...
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### Indefinite Integral

I tried to solve this indefinite integral $$\int\frac{1}{1+\tan^{-1}x}dx$$ I try taking the change of variable $u=\tan^{-1}x$ but I fail to reach a solution. Can anyone help me. Thanks in advance.
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### Definite integral including the Chebyshev polynomial

I would like to know the proof of $$\int_a^b \frac{T_n(x/a)T_n(x/b)\, dx}{x(b^2-x^2)^{1/2}(x^2-a^2)^{1/2}}=\frac{\pi}{2 ab}, 0<a<b, n \in \Bbb N$$ where $T_n(x)$ is the Chebyshev polynomial of ...
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### Is this integral expressible in terms of generalized hypergeometric functions?

While carrying out a calculation, I encountered this integral: $$\int_0^1 d u~u^{-1-2 x} (1-u)^{-x} \Big({}_2F_1\left(1, 1, 1-x; u\right)\Big)^2\,.$$ I read in Exton's book that it is expressible in ...
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### Determine the behavior of a function defined by an integral

Suppose we have a function defined by $$\varphi(s)=\int_{-\infty}^\infty f(x,s)\,dx$$ defined for $s\in S\subseteq \mathbb{R}$. Suppose we know that it blows up at $a\in \partial S$, and we want to ...
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### infinite sum and integral representation

assuming all the numbers $k_{n}$ are REAL is then true that $$\sum_{n=0}^{\infty}\frac{1}{k_{n}^{s}}=\frac{1}{2 \Gamma (s) cos(\pi s)}\int_{0}^{\infty}dtt^{s-1}\sum_{n=0}^{\infty}cos(k_{n}t)$$ ...
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