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Evaluate \(\displaystyle{\int_{0}^{1}\tan^{-1}(x)\ln(x)\,\mathrm{d}x}\). If the answer can be expressed as ...

\[\sum_{\mu(n)=1} \frac{1}{n^2} = \frac{A}{B\pi^C}\]

Let \(\mu(n)\) denote the möbius function, the sum is taken over all positive integers \(n\) such ...

\[\sum_{n=1}^{\infty} \dfrac{\zeta(3n) - 1}{n} = \ln(A \pi) + \pi \cos\left(\dfrac{\pi}{B}\right) - \ln\left(e^{\pi \sqrt{C}} + \tan\left(\dfrac{\pi}{D}\right)\right)\]

You have recently bought the Random Number Generator 4000 from your local hardware store. This remarkable machine outputs numbers randomly and uniformly in the range \([0, 1]\). You decide to ...

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