Formelsammlung Mathematik: Bestimmte Integrale: Form R(x,cos,arccos)

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Coxeter Integrale

0.1

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\arccos \left(\frac{\cos x}{1+2\cos x}\right)dx=\frac{5{\pi }^2}{24}$

0.2

$\underset{0}{\overset{\frac{\pi }{3}}{\int }}\arccos \left(\frac{\cos x}{1+2\cos x}\right)dx=\frac{2{\pi }^2}{15}$

0.3

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\arccos \left(\frac{1}{1+2\cos x}\right)dx=\frac{{\pi }^2}{6}$

0.4

$\underset{0}{\overset{\frac{\pi }{3}}{\int }}\arccos \left(\frac{1}{1+2\cos x}\right)dx=\frac{{\pi }^2}{8}$

0.5

$\underset{0}{\overset{\arccos \left(\frac{1}{3}\right)}{\int }}\arccos \left(\frac{1-\cos x}{2\cos x}\right)dx=\frac{{\pi }^2}{6}$

0.6

$\underset{0}{\overset{\frac{\pi }{3}}{\int }}\arccos \left(\frac{1-\cos x}{2\cos x}\right)dx=\frac{11{\pi }^2}{72}$

0.7

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\arccos \sqrt{\frac{\cos x}{1+2\cos x}}dx=\frac{{\pi }^2}{6}$

0.8

$\underset{0}{\overset{\frac{\pi }{3}}{\int }}\arccos \sqrt{\frac{\cos x}{1+2\cos x}}dx=\frac{5{\pi }^2}{48}$