# 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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### Rewrite order of $\int_0^1\int_0^{\sqrt{y}} \int_y^1 \, dz \, dx \, dy$ to $dx\,dy\,dz$ and $dy\,dz\,dx$

I need to change the order of $$\int_0^1\int_0^{\sqrt{y}}\int_y^1\,dz\,dx\,dy$$ to $dx\,dy\,dz$ and $dy\,dz\,dx.$ I can extract the inequalities to get $1≤z≤y$, $0≤x≤\sqrt y$, $0≤y≤1$, but I get ...
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### Changing the order of integration in three dimensions, of a level curve, $e^{x^2}$

Change the order of integration $dx\,dy$ to $dy\,dx$ and evaluate it: $$\int_0^1\int_{3y}^3 e^{x^2} \, dx \, dy$$ now I now that I can do the following: $$x=3,\quad x=3y \to y=\frac{1}{3}x$$ Okay ...
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### How to integrate $\int \dfrac{1}{\sin^4 x \cos^4 x} dx$

The integral in question is: $$\int \dfrac{1}{\sin^4 x \cos^4 x} dx$$ I tried using $1 = \sin^2 x + \cos^2 x$, but it takes me nowhere. Another try was converting it into $\sec$ and $\csc$, but ...
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### Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1}$

As the question says, $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1}$$ where a is a constant, $a>0$.
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### Area of circle (double integral and cartesian coordinates)?

I know that the area of a circle, $x^2+y^2=a^2$, in cylindrical coordinates is $$\int\limits_{0}^{2\pi} \int\limits_{0}^{a} r \, dr \, d\theta = \pi a^2$$ But how can find the same result with a ...
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### $\int\frac{x }{ \sqrt {3-x^4}}{dx}$ by substitution method

I am trying to determine via substitution: $$\int\frac{x }{ \sqrt {3-x^4}}{dx}$$ My work: $$x=\frac{1}{t}$$ $$dx=-\frac{dt}{t^2}$$ $$- \int\frac{dt }{ t \sqrt {3t^4-1}}$$ How to proceed ...
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### Conjectured value of $\int_{0}^{\infty}\left(\frac{x-1}{\ln^2 x}-\frac{1}{\ln x}\right)\frac{\mathrm{d}x}{x^2+1}$

I was curious whether this integral has a closed form expression : $$\int_{0}^{\infty}\left(\frac{x-1}{\ln^2 x}-\frac{1}{\ln x}\right)\frac{\mathrm{d}x}{x^2+1}$$ The integrand has a singularity ...
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### how to partial fraction $\frac{1}{(x+1)^2}$

I need to integrate $\frac{1}{(x^2+2x+1)}$, so I need to use partial fraction as the polynomial can be factored as $\frac{1}{(x+1)^2}$. This is what I've tried: $$\frac{A}{(x+1)} + \frac{B}{(x+1)^2}$$...
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### Is an average function integrable?

I'm thinking about the following question: If $u\in L^p(\mathbb{R}^n)$, is $f(x)=\int_{|y-x|<R}|u(x)-u(y)|^pdy$ in $L^1(\mathbb{R}^n)$, where $R>0$ is a fixed numbers? It's clear that if $u$ ...
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### Change the order of integral

How to change the integral $\int_a^y G(x)\int_a^x F(t) \int_a^t W(z) dzdtdx$ into the form $\int_{?}^{?} F(x) \int_a^{?} G(t) \int_a^{?} W(z)dzdtdx$
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### $\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty$ implies $\{f_n\}$ is uniformly integrable?

Suppose $\mu$ is a finite measure and for some $\gamma > 0$, we have$$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty.$$Does it follow that $\{f_n\}$ is uniformly integrable?
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### How solve $\int_{0}^{\infty} \dfrac{1-\cos x}{x^{2}} dx$ [closed]

What is the value of the following integral? $$\int_{0}^{\infty} \dfrac{1-\cos x}{x^{2}} dx$$
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### Proving that an infinite series equal a finite series

Suppose we have a function $f(z)$, which has $m$ isolated singularities, which are non-integers (say, $z_1$, $z_2$,...,$z_m$). Define $H(z):=\frac{\pi f(z)}{\sin(\pi z)}$. Assume that there exists a ...
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### Does it follow that $\{f_n\}$ is uniformly integrable?

Suppose $\mu$ is a finite measure, $f_n \to f$ almost everywhere, each $f_n$ is integrable, $f$ is integrable, and $\int |f_n - f| \to 0$. Does it follow that $\{f_n\}$ is uniformly integrable?
### Integral of the level curve $ye^{2x}$
$\int_{0}^{2}\int_{0}^{4} ye^{2x}$ $dydx$ The book says the answer is $32(e^4-1)$ I had to do u-substitution when doing it, maybe that's where I went wrong. First integrating with respect to y, ...