Tagged Questions

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Geometric interpretation of an integral inequality

Let $f: [a, b] \to \mathbb [0, \infty)$ be an integrable function. By Cauchy-Schwartz: $$\left(\int_a^b f(x) dx\right)^2 \leq (b-a) \int_a^b f(x)^2 dx$$ with equality iff $f$ is constant. If we ...
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What is the value of the unknown parameter so that the given area condition holds?

The graphs of $f(x) \colon= x^2$ and $g(x) \colon= cx^3$, where $c > 0$, intersect at the points $(0,0)$ and $(1/c, 1/c^2)$. What is the value of $c$---and how to compute this value---so that the ...
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Definite Integral theorem validity :- $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds$?

Can we write $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds\tag 1$ ? In other words, is this result valid? If so, could you help me to get the proof it NB :: ...
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For what $p$ does this series converge

"Find the values of $p$ s.t. the following series converges: $\sum_{n=2}^{\infty} \frac{1}{n^p \ln(n)}$" I am trying to do this problem through using the Integral Test to find the values of $p$. I ...
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How to find ${\large\int}_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$$
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Lower bound on $F$ under the assumption $\theta F(s)\le sF'(s)$

Let $F(s)=\displaystyle \int_0^{s}f(t)\,\mathrm dt$. We suppose that there exists $\theta>2$ such that $\theta F(s)\le f(s)s$ for all $s\in \mathbb{R}$ and that $F(s)>0$ for all ...
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Volume and surface area of elliptic torus

ok, so i have an elliptic torus, parametrised as (i'll just copy the mathematica syntax): rr := {(R + a Cos[v]) Cos[u], (R + a Cos[v]) Sin[u], b Sin[v]} R is here the 'big' radius, and a and b are the ...
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Explanation of the passage from $\int_{N'}^N dN/N$ to $\ln N-\ln N'$

While going through my text I got stuck in the derivation given in the picture. ($\Omega$ is a constant) I don't know how to get the second step from the first step, also I don't know why ln is ...
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Proving that $\displaystyle \int_{0}^{a} f(x) \;\mathrm dx = \int_{0}^{a} f(a - x) \;\mathrm dx$

The question I have is: Prove that $\displaystyle \int_{0}^{a} f(x) \; \mathrm dx = \int_{0}^{a} f(a - x) \; \mathrm dx$ Since this question occurs at the end of an exercise on integration by ...
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Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
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Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) ...
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Power series function expansion as solution for integral equation

I'm facing an integral equation whose unknown is a function $f(x)$: The equation is of the kind: $$K = \int_{-l}^{l} G(x,s)f(s)ds$$ So it's a Fredholm integral equation that is rewritten in this ...
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Spectral density of a sample covariance matrix in a Gaussian Random Ensemble

Let $N > 0$ and $T > N$ be integers and $C$ be a real, symmetric $N \times N$ matrix.The question is to compute the following integral: U(t) := \frac{1}{N} ...
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General form for $2\int_{0}^{\infty} \frac{1-t^2}{(1+t^2)((a+b)t^2+a-b)} \mathrm dt$
I encountered this integral in physics-- $$2\int_{0}^{\infty} \dfrac{1-t^2}{(1+t^2)((a+b)t^2+a-b)} \mathrm dt$$ I know for certain that $a>0$, $b>0$. $a$ and $b$ are independent variables