# 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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### Volume of $\Gamma=\{(x,y,z)\in \mathbb{R}^3:\sqrt{x^2+y^2}\le z\le \sqrt{2x^2+2y^2};x^2+y^2+z^2\le 3\}$

I evaluated the volume of the set $$\Gamma=\{(x,y,z)\in \mathbb{R}^3:\sqrt{x^2+y^2}\le z\le \sqrt{2x^2+2y^2};x^2+y^2+z^2\le 3\}$$ by using the Pappu's centroid theorem, but I'm in trouble while ...
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### Integrate $\int_0^T \frac{e^s (1+\beta e^{\theta s})}{\sqrt{1+e^s (1+\beta e^{\theta s})} }\, ds$

Integrate $$\int_0^T \frac{e^s (1+\beta e^{\theta s})}{\sqrt{1+e^s (1+\beta e^{\theta s})} }\, ds$$ I have tried integration by parts and change of variable but I'm not able to solve thus far. From ...
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### Find the volume of the solid formed by the revolving the region around a line

I have to find the volume of the solid formed by the revolving the region enclosed by $x=\frac{y^2}4$ and $y=x^5+x^3$ around the line $y=2$ I know how to find the volume when it revolve around x-axis,...
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### Convergence of Fourier series at $x=0$

Let $f$, $2\pi$-periodic and intergrable function defined as follows: $$f(x) = \begin{cases} 1+\sin\frac{\pi^2}{x} & x\in[-\pi,\pi),x\ne 0 \\ 1 & x=0 \end{cases}$$ Does the Fourier ...
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### Improper integral of $\log x \operatorname{sech} x$

How to prove the following? $$\int_0^\infty \log x \operatorname{sech}x\,dx = \frac{\pi}{2} \log\left( \frac{4\pi^3}{\Gamma(1/4)^4} \right)$$ I obtained the right side with CAS. It seems like this ...
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### Lebesgue integrable and improper Riemann integrable

I'm sitting here on a task, where I have to show that for the function: $$f:(0,1]\times(0,1]\to\mathbb R, \quad f(x,y)=\frac{x-y}{(x+y)^3},$$ the double Riemann-integral is the double Lebesgue-...
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### Change of variables for the operator curl

I am working on a finite element problem and I have some difficulties. Until now I have worked with gradients and I know how to work with this operator. But now I have to work with the curl operator ...
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### Compute $\int_n^\infty \frac{\text{d}x}{x^n+\sin(nx)}$

$$\int_n^\infty \frac{\text{d}x}{x^n+\sin(nx)}$$ (In terms of $n$) I've struggled on this for quite some time because it seems to prove impossible to break up that extremely annoying $\sin (nx)$ term....
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### Integrals using substitution $x\mapsto x+\frac{1}{x}$

Do you have some links to questions with integrals that require the substitution $x\mapsto x+\frac{1}{x}$ (as a trick)? I know this trick, but I can't seem to find any integrals to practice it..
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### Integral over $[0,1]$ of $g\left(x^{1/(2n+1)}\right)$ vanishes

Suppose $g:[0,1] \to \mathbb{R}$ is continuous and such that $$\int_0^1 g\left(x^{\frac{1}{2n+1}}\right)\,dx = 0, \quad n=0, 1, 2, \dots$$ Show that $g \equiv 0$ on $[0,1]$. My idea was to use ...
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### Evaluate $\int_{C}\frac{e^{3z}}{(z+1)^5}dz$

$$\int_{C}\frac{e^{3z}}{(z+2)^5}dz\,\,\,\,\,\,\,\,C:|z|=3$$ My try: Applaying Cauchy's formula $$\int_{C}\frac{e^{3z}}{(z+2)^5}dz=2\pi ie^{-2\cdot 3}=\frac{2\pi i}{e^{6}}$$ I'm not sure in my ...
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### How to prove that $\int_0^\infty\frac{\left(x^2+x+\frac{1}{12}\right)e^{-x}}{\left(x^2+x+\frac{3}{4}\right)^3\sqrt{x}}\ dx=\frac{2\sqrt{\pi}}{9}$?

A friend gave me this integral as a challenge $$\int_0^\infty\frac{\left(x^2+x+\frac{1}{12}\right)e^{-x}}{\left(x^2+x+\frac{3}{4}\right)^3\sqrt{x}}\ dx=\frac{2\sqrt{\pi}}{9}.$$ This integral can be ...
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### Evaluating $\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x$

I need to evaluate $$\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x$$ I've been told that the way forward is ...
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### Converting cartesian double integral into polar integral

Write down the integral $$\int_0^4 \int_x^ {4x} \sqrt{x^2+y^2}dydx$$ $(i)$ integration with respect to $\theta$ first and $r$ second.
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### Find the volume between two paraboloids

Find the volume of the solid enclosed by the paraboloids $z = 1-x^2-y^2$ and $z = -1 + (x-1)^2 + y^2$. Using triple integrals, it is known that $V = \iiint_R \mathrm dx\,\mathrm dy\,\mathrm dz$, and ...
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### Inverse short time Fourier transform

The short time Fourier transform $S: L^2(\mathbb{R})^2 \rightarrow L^2(\mathbb{R}^2)$ can be defined as $$S(g,f)(a,b):=\int_{\mathbb{R}}f(x) \overline{g(x-a)} e^{-i b x} dx.$$ Now a natural question ...
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### Integration for radial equations: $\int \limits _0 ^r \big( 2u-Au^2 \big) e^{-Au} du$

I have a Physics problem however I have an issue with an integration midway through my question. I have $$\int \limits _0 ^r \big( 2u-Au^2 \big) e^{-Au} du .$$ Where A is a constant I have tried ...
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### Differential Equations (Proving)

This is the question --> What I have done; $$dS/dt = kS(N-S)$$ $$1/(S(N-S)) = k dt$$ $$1/(SN - S^2) = k dt$$ Therefore $$1/(S-2S) * ln|SN-S^2| = kt + c$$ A I on the right track? ...
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### How do I calculate $\int_0^ax^2\sqrt{a^2-x^2}dx$ via substitution?

I have to calculate the integral $$\int_0^ax^2\sqrt{a^2-x^2}dx$$ using solely substitution (no integration by parts). $a$ is a positive constant. I'm confused on how to do this?
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### complex integration using residue theorem

I have the following integration $$g_n=\frac{(-1)^n}{2\pi} \int_{-\infty} ^ {\infty} (\frac {e^{-itx}}{(it-1)^n})dt$$. I used the residue theorem to find the integration so the result of the ...
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### Solve the double integral $\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)dxdy\:$ [closed]

$$\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)\,dx\,dy.$$ I think you need to be solved by the transition to polar coordinates: \begin{cases} x=r\cos(\phi),\\ y=r\sin(\...
Let $f:\mathbb{R}\mapsto\mathbb{R}$ be a continuous function such that $f(x)\geq 0$ for all $x$ and \begin{align*} \int_{-\infty}^{\infty}f(x) = 1 \end{align*} For $r\geq 0$, let \begin{align*} I_{...
Find $\displaystyle \int \sqrt{\tan(x)}dx$. According to an integral calculator, the answer to this question is (-2 \tan^{-1}(1-\sqrt(2) \sqrt(\tan(x)))+2 \tan^{-1}(\sqrt(2) \sqrt(\tan(x))+1)+\log(...