# 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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### How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?

Calculate $$\iiint_{B} y\;dxdydz.$$ The set is $\;B=\{(x,y,z) \in \mathbb R^3$; $\; x^2+y^2+4z^2\le12$, $-x^2+y^2+4z^2\le6$, $y\ge 0 \}$. I know that B is defined by a real ellipsoid, an ...
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### Volume of the $N$-dimensional domain $\sum\limits_{k=1}^N (1 + |x_k|^a)^b\le\epsilon$

I wish to calculate the following $N$-dimensional integral $$I = \int_0^\infty dx_1 \ldots \int_0^\infty dx_{N} \, H\left(\epsilon - \sum_{k=1}^N (1 + x_k^a)^b\right),$$ where $a, b$ and $\epsilon$ ...
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### Is there a reduction formula for $I_n=\displaystyle\int_{0}^{n\pi}\frac{\sin x}{1+x}\,dx$?

I haven't been able to manipulate this integral. I need to find the value of $I_n$ for $n=1,2,3,4$ and arrange them in ascending order.
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### Correlation of an integral

Assuming that $a(x)$ is a random variable and correlation $\xi(x_1,x_2)=\langle a(x_1)a(x_2)\rangle$ is known (where angular bracket denote statistical averaging), it it possible to write the ...
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### $\lim_{n \to \infty}\int_{0}^{a} e^{nx} dx$

Find $\lim_{n \to \infty}\int_{0}^{a} e^{nx} dx$ This seems to be a straightforward problem but since I am new to defnite integrals and thishas appeared in one the graduate exam papers I am looking ...
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### Simplify an integral or the integrand involving hyperbolic functions

I would like to simplify the following integral or at least the integrand: $$f(t):=\int_{a}^{t-a} \frac{(\cosh(t-x)-\cosh(a))^{i\tau-1/2}}{(\cosh(x)-\cosh(a) )^{i\tau+1/2}} dx, \quad t>2a,$$ ...
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### Why does Maple include x in the solution of this definite integral?

I have the following function defined in Maple: $$f(x) := (2 - a + ax^2) \sqrt{1 + 4a^2x^2}$$ And I want to calculate the definite integral of this from -1 to 1: $$\int_{-1}^{1}{f(x)dx}$$ I do ...
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### How to calculate this integral $\int_0^a {dx\frac{{\tanh \left( x \right)}}{x}}$?

I encountered this integral, which Mathematica can't give an answer. $$\int_0^a {dx\frac{{\tanh \left( x \right)}}{x}}$$ I am sure the result contains Euler constant. How to do it?
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### Calculate the area of a sphere drilled by two cylinders.

Let $S$ be the sphere given by the equation $x^2+y^2 +z^2 =4$ cut with $z \geq 0$. Now, we drill the semisphere that is left with two vertical cylinders of radius $1$, whose axes are respectively ...
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### Integration in Maple 16

I need to solve this integral in Maple 16, I've tried various functions such as evalf, solve, ...
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### Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives

I am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old series,...
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### Intuition/derivation for Cauchy's repeated integral formula?

https://en.m.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration I'm referring to this formula due to Cauchy. The wiki page has a proof, but what I'm looking for is a more direct derivation or ...
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### Reference: Gaussianity of linear functional of Gaussian process

My question is similar to this one, but I'm looking for a reference rather than derivation. I've been told, inserting my own commentary in square brackets, If you take $X$ in $C([a,b])$ [i.e., $X$...
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### Integrable and primitivable functions

Recently we learned the fundamental theorem of calculus and subsequently Leibniz-Newton formula, that linked the primitive with the definite integral. I know that Leibniz-Newton can be applied when ...
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### What is $\int x\tan(x)dx$?

I have a problem when trying to solve this question Question. What is the answer of the indefinite integral $$\int x\tan x \; dx?$$ Maple gives a complicated answer based on the series. Is there any ...
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### Calculating the Stokes Theorem

I was tasked with calculating $\oint_{L}Fdr$ for when $F=xzi-j+yk$ (vetor form) and $$L = \begin{cases}z=5(x^2+y^2)-1 & \mbox{ } \mbox{} \\z=4 & \mbox{} \mbox{} \end{cases}$$ Using: ...
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### How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}}$?
How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}}$ ? I tried substituting $x=1/t$ but that's making it more complicated.Any suggestions?