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Integral

Calculus (Integrals, Series)
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Riemann
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Integral

#1

Post by Riemann » Tue Feb 08, 2022 9:31 pm

Let $a\geq 1$. Prove that:

$$\int_{0}^{\pi} \left ( 1 + \cos x \right ) \ln \left ( a + \cos x \right )\, \mathrm{d}x = \pi \left ( a - \sqrt{a^2-1} \right ) + \pi \ln \left ( \frac{a + \sqrt{a^2-1}}{2} \right )$$
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$

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