Formelsammlung Mathematik: Bestimmte Integrale: Form R(x,log)

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0.1

$\underset{0}{\overset{1}{\int }}\frac{\log (1+x)-\log 2}{1+x^2}dx=-\frac{\pi }{8}\log 2$

0.2

$\underset{0}{\overset{1}{\int }}\frac{\log (1+x)-\log 2}{1-x^2}dx=-\frac{{\pi }^2}{24}$

0.3

$\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\log (1+x)-\log 2}{1+x^2}dx=G-\frac{\pi }{8}\log 2$

0.4

$\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\log (1+x)-\log 2}{1-x^2}dx=-\frac{{\pi }^2}{12}$

0.5

$\underset{0}{\overset{1}{\int }}\frac{\log x}{1+x^2}dx=-G$

0.6

$\underset{0}{\overset{1}{\int }}\frac{\log x}{1-x^2}dx=-\frac{{\pi }^2}{8}$

0.7

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\log (1+x+x^2)}{1+x^2}dx=\frac{\pi }{3}\log (2+\sqrt{3})+\frac{4}{3}G$

0.8

$\underset{0}{\overset{1}{\int }}\log (-\log x)dx=-\gamma$

0.9

$\underset{0}{\overset{1}{\int }}(\frac{2\log x}{x^2-4x+8}-\frac{3\log x}{x^2+2x+2})dx=G$

0.10

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\log (x+1)}{{\log }^{2}x+{\pi }^2}\frac{dx}{x^2}=\gamma$

0.11

$\underset{0}{\overset{1}{\int }}\frac{\log (1+x^{2+\sqrt{3}})}{1+x}dx=\frac{{\pi }^2}{12}\cdot (1-\sqrt{3})+\log (2)\cdot \log (1+\sqrt{3})$

0.12

$\underset{0}{\overset{1}{\int }}\frac{\log (1+x^{4+\sqrt{15}})}{1+x}dx=\frac{{\pi }^2}{12}\cdot (2-\sqrt{15})+\log \left(\frac{1+\sqrt{5}}{2}\right)\cdot \log (2+\sqrt{3})+\log (2)\cdot \log (\sqrt{3}+\sqrt{5})$

0.13

$\underset{0}{\overset{1}{\int }}\frac{\log (1+x^{6+\sqrt{35}})}{1+x}dx=\frac{{\pi }^2}{12}\cdot (3-\sqrt{35})+\log \left(\frac{1+\sqrt{5}}{2}\right)\cdot \log (8+3\sqrt{7})+\log (2)\cdot \log (\sqrt{5}+\sqrt{7})$

0.14

$\underset{0}{\overset{1}{\int }}\log (1-x)\log (x)\log (1+x)dx=-6+\frac{5{\pi }^2}{12}-(\log 2{)}^2+4\cdot \log 2-\frac{{\pi }^2}{2}\log 2+\frac{21}{8}\zeta (3)$

0.15

$\underset{0}{\overset{1}{\int }}\frac{\log (1+x)\log x}{1+x}dx=-\frac{\zeta (3)}{8}$

0.16

$\underset{0}{\overset{1}{\int }}\frac{\log (1+x)\log x}{x}dx=-\frac{3}{4}\zeta (3)$

0.17

$\underset{0}{\overset{1}{\int }}\frac{\log (1+x)\log x}{1-x}dx=-\frac{{\pi }^2}{4}\log 2+\zeta (3)$

0.18

$\underset{0}{\overset{1}{\int }}\frac{\log (1-x)\log x}{1+x}dx=-\frac{{\pi }^2}{4}\log 2+\frac{13}{8}\zeta (3)$

0.19

$\underset{0}{\overset{1}{\int }}\frac{\log (1-x)\log x}{x}dx=\zeta (3)$

0.20

$\underset{0}{\overset{1}{\int }}\frac{\log (1-x)\log x}{1-x}dx=\zeta (3)$

0.21

$\underset{0}{\overset{1}{\int }}\frac{\log (1-x)\log (1+x)}{x}dx=-\frac{5}{8}\zeta (3)$

0.22

$\underset{0}{\overset{1}{\int }}\frac{\log (1-x)\log (1+x)}{1+x}dx=\frac{(\log 2{)}^2}{8}-\frac{{\pi }^2}{12}\log 2+\frac{\zeta (3)}{8}$

1.1

$\underset{0}{\overset{1}{\int }}\frac{{\log }^{2n}x}{1+x^2}dx=\frac{1}{2}|E_{2n}|{\left(\frac{\pi }{2}\right)}^{2n+1}n\in {\mathbb{Z}}^{\ge 0}$

1.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{{\log }^{n-1}x}{1+x^2}dx=|E_{n-1}|{\left(\frac{\pi }{2}\right)}^{n}n\in {\mathbb{Z}}^{\ge 1}$

1.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{{\log }^{n-1}x}{1-x^2}dx=\frac{2^n(1-2^n)|B_n|}{n}{\left(\frac{\pi }{2}\right)}^{n}n\in {\mathbb{Z}}^{\ge 1}$

1.4

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{dx}{(x+1{)}^n({\log }^{2}x+{\pi }^2)}=C_n$ Fontana-Zahlen genügen der Rekursion: $C_0=-1,\sum _{k=0}^{n-1}\frac{C_k}{n-k}=0$

1.5

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{1}{x+1-u}\cdot \frac{1}{{\log }^{2}x+{\pi }^2}dx=\frac{1}{u}+\frac{1}{\log (1-u)}u\in {ℂ}^{\times }\setminus {ℝ}^{\ge 1}$

1.6

$\underset{0}{\overset{1}{\int }}\frac{{(\log \frac{1}{x})}^{z-1}}{1+x}dx=\eta (z)\mathrm{\Gamma }(z)\text{Re}(z)>0$

1.7

$\underset{0}{\overset{1}{\int }}\frac{{(\log \frac{1}{x})}^{z-1}}{1-x}dx=\zeta (z)\mathrm{\Gamma }(z)\text{Re}(z)>1$

1.8

$\underset{0}{\overset{1}{\int }}\frac{x^{\alpha -1}-x^{-\alpha }}{(x+1)\log x}dx=\log (\tan \frac{\alpha \pi }{2})0<\mathrm{R}\mathrm{e}(\alpha )<1$

1.9

$\underset{0}{\overset{1}{\int }}{(\log \frac{1}{x})}^{z-1}dx=\mathrm{\Gamma }(z)\text{Re}(z)>0$

1.10

$\underset{0}{\overset{1}{\int }}(\frac{\log x}{a+1-x}-\frac{\log x}{a+x})dx=\frac{1}{2}{(\log a-\log (a+1))}^2\mathrm{\forall }a\in ℂ\setminus [-1,0]$

1.11

$\underset{0}{\overset{1}{\int }}\frac{2\log x}{(a\pi {)}^2+{\log }^{2}x}\frac{x}{1-x^2}dx=\frac{1}{2a}+\psi (a)-\log a\text{Re}(a)>0$

1.12

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\log (1+2\sin \alpha x+x^2)}{1+x^2}dx=\pi \log (2\cos \frac{\alpha }{2})+\alpha \log (\tan \frac{\alpha }{2})+2\sum _{k=0}^{\mathrm{\infty }}\frac{\sin (2k+1)\alpha }{(2k+1{)}^2}0<\alpha <\pi$

1.13

$\underset{0}{\overset{1}{\int }}\frac{\log \log \left(\frac{1}{x}\right)}{1+2\cos \alpha \pi \cdot x+x^2}dx=\frac{\pi }{2\sin \alpha \pi }(\alpha \log 2\pi +\log \frac{\mathrm{\Gamma }(\frac{1}{2}+\frac{\alpha }{2})}{\mathrm{\Gamma }(\frac{1}{2}-\frac{\alpha }{2})})0<\alpha <1$

1.14

$\underset{0}{\overset{1}{\int }}\frac{\log (1-x)}{x}\frac{2z}{{\log }^{2}x+(2\pi z{)}^2}dx=-\log \left(\frac{z!e^z}{z^z\sqrt{2\pi z}}\right)\text{Re}(z)>0$

2.1

$\underset{a}{\overset{b}{\int }}\frac{\log x}{(x+a)(x+b)}dx=\frac{\log (ab)}{2(b-a)}\log \left(\frac{(a+b{)}^2}{4ab}\right)$

2.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\log x}{(x+a)(x+b)}dx=\frac{{\log }^2(a)-{\log }^2(b)}{2(a-b)}$