# 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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### Find $\lim_{n \to \infty} n \int_0^1 (\cos x - \sin x)^n dx$

Find: $$\lim_{n \to \infty} n \int_0^1 (\cos x - \sin x)^n dx$$ This is one of the problems i have to solve so that i could join college. I tried using integration by parts, i tried using ...
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### Divergence Theorem: Conditions for the boundary integration to vanish?

Consider the Divergence Theorem for example in two dimensions, in the upper right quadrant of Euclidean space: $$\int_0^\infty dx \int_0^\infty dy ~\vec\nabla\cdot\vec F=\oint_C ds~\vec n\cdot\vec F$$...
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### Differentiation under the integral sign in $R^3$

I'm trying to take derivative from an integral. I know about the Reynolds transport theorem, but I do not know how to obtain the unit normal and the velocity. I'm going to take the derivate from the ...
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### Certain type of integrals $\int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}},$

I would like to do the following integral $$\int_0^{\pi}d\theta\sin\theta \frac{1}{x+i\epsilon - \sqrt{y+z\cos\theta}},$$ where the $i\epsilon$ has been added to avoid some possible divergencies. ...
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### Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
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### Integral of a line with random gradient

Consider a random line $Y = Mx$ where $M$ is a standard normal variable $M \sim \mathcal{N}(0,1)$. The line is integrated between 0 and 1: $$I = \int_{0}^{1} Y dx = \int_{0}^{1} Mx dx$$ What is the ...
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### Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}xdx$ equal to zero?

My initial question was to find if this integral $$\int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
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### double integration changing variables and finding boundaries

This may be quite simple, but I am a little rusty and have had quite a few attempts at this and would appreciate some help. The form of the integrand does not really matter, I am trying to change the ...
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### Integral of following fractional part

What is the integral of \int {nx} where {.} is a fractional part and $n\in R$ the integration range is from $-4$ to $4$.I want to use it in one of problems of the ellipse. Thanks . I haven't shown any ...
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### $\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin y}{ny(1+n^2y^2)}ndy$ via DCT?

I'm looking to calculate these limits/integrals: $$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin (x/n)}{x(1+x^2)}dx$$ 2.$$\lim_{n\to\infty}\int_0^{\infty}\dfrac{\sin(x/n)}{(1+x/n)^n}dx$$ I posted ...
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### Good books on integrals [duplicate]

I'm a math student at the sixth semester and I've had my courses in calculus and complex analysis. I'm able to solve integrals with the usual techniques. However, whenever I am confronted with a ...
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### Understanding how to calculate surface area of parametrized surfaces

I am trying to follow a derivation for surface area of a parameterized surface and my book does not explain the reasoning behind different steps. I understand the derivation for surface area for a ...
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### Integral of combination of power, exponential, and confluent hypergeometric function

I am trying to solve a couple integrals of the form: \int_{0}^{\infty} x \, e^{-a(gx-b)^{2}}\,e^{-\beta_{1}x}\, {_{1}}F_{1}(-\alpha_{1};-\alpha_{3};\beta_{3} x) \ \mathrm{d}x \end{...
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### Integrated Brownian Bridge is a Gaussian Process

Let $W(t),t \in [0,1]$ be a (Standard) Wiener Process. The Brownian Bridge $B(t), t \in [0,1]$ can be constructed via $B(t):=W(t) - t \cdot W(1)$ and is a Gaussian process with zero mean and ...
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### Does make sense define a gauge for the integral $\int_2^x\frac{\sum_{n\leq t}\Lambda(n)}{t}dt$, where $\Lambda(n)$ is the von Mangoldt funtion?

I try encourage to me to study and understand the definition of gauge integral. See for example this reference Schechter, The Gauge integral where is explained the definition with an example. It is ...
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### Evaluate $\int \frac{x}{x^4+5}dx$

$$\int \frac{x}{x^4+5}dx$$ $u=x^2$ $du=2xdx\Rightarrow \frac{du}{2}=xdx$ $$\int \frac{x}{x^4+5}=\frac{1}{2}\int \frac{du}{u^2+5}$$ I want to get to the expression in the form of $\frac{da}{a^2+1}$...
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### how is it that $\int_0^x 2\pi r\ dr$ is equal to the area of a circle [closed]

I'm studying calculus and I'm having some basic questions, this one is regarding the area of a circle. we know, from some guy, that the circumference of a circle is $2 \pi r$ and the area can be seen ...
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### Dominance between two functions

Let two functions $f(z)$ and $g(z)$ with $z\in[0,c]$ with $c$ a constant such that $c<b$. I'd like to check whether $f(z)-g(z)>0$. I've tried to set $f(z)$ to its minimal value and $g(z)$ to its ...
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### Centroid of circle intersection

A region $R$ is bounded by the circle radius $4$ centred at $(5,0)$, the circle radius $2$ centred at $(0,0)$ and the x-axis, as shown. What is the centroid of this region? Finding the top ...
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### Integral of product of a sigmoid function and beta distribution over (0, 1)

I am trying to solve the following integral, $$\int_0^1 \frac{1}{1+e^{-x}}\cdot Beta(x|\alpha, \beta)dx$$ where $Beta(x|\alpha, \beta)=\frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}$ is a ...
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### Method of integration [duplicate]

We have to find the integration of the following function I tried but got stuck can anybody help me how to proceed . Is there anyother method to solve this
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### Integrals in Reverse

I'm asked what solid this integral represents(the integral is used to obtain the volume of a solid, we are given this). I see that since we have the $2\pi$, this is probably a volume obtained by ...
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### Indefinite Integral of $\frac{1}{1+x^8}$ [closed]

How to find the indefinite Integral of $\dfrac{1}{1+x^8}$ by hand? Thanks for help. Please show the steps and the method used.
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### Integration problems The curve in the picture shown has equation y = bx(x − 2)… [closed]

I really need urgent help for this and I would like to thank people for your help in advance. Thank you. The problem set is here. Just click it! Integration
Let f be an integrable real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by F(x)=$\int _{a}^{x}\!f(t)\,dt$ Doesn't this make F(a) = $0$ ?
I'm currently operating with the following integral: $$\int\frac{u'(t)}{(1-u(t))^2} dt$$ But I notice that $$\frac{d}{dt} \frac{u(t)}{1-u(t)} = \frac{u'(t)}{(1-u(t))^2}$$ and \frac{d}{dt} \frac{...