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Find the minimum number of nodes \(n\) on a graph, that satisfies the following properties:

\[\large V =\int_1^2 \int_1^2 \int_1^2 f(x,y,z) \ln \left( 1 + \frac{1}{1 + f(x,y,z)} \right) \text{ d}x \text{ d}y \text{ d}z \]

If

\[\int\limits_0^1 \frac{(\ln x)^2}{x-1}\left[ x^{-1/3} - x^{-2/3} \right]\,\mathrm dx\]

can be expressed in the form ...

\[ \large f(x) = \frac2{e^x - e^{-x}} \left( 1 + \int_1^x f(t) \, dt \right) \]

Suppose a function \(f\) defined on \(x>0\) satisfy the equation above, find the value ...

\[\large{ S = \sum_{n=1}^\infty \dfrac{a_{2n+2}}{a^2_{n-1} a^2_{n+1}} }\]

Let \((a_n)\) be a sequence defined by \(a_0 = 1, a_1 = 2\), and for ...

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