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

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0.1

$\underset{0}{\overset{1}{\int }}\frac{\arctan x}{x}dx=G$

0.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\arctan x}{1+x^2}dx=\frac{{\pi }^2}{8}$

0.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\arctan x}{1-x^2}dx=-G$

0.4

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x\arctan x}{1+x^4}dx=\frac{{\pi }^2}{16}$

0.5

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x\arctan x}{1-x^4}dx=-\frac{\pi }{8}\log 2$

0.6

$\underset{0}{\overset{1}{\int }}\frac{\arctan \left(x^{3+\sqrt{8}}\right)}{1+x^2}dx=\frac{1}{8}\log (2)\cdot \log (1+\sqrt{2})$

1.1

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\arctan ax}{x(1+x^2)}dx=\pi \log (1+a)a\ge 0$

1.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\arctan ax}{x(1-x^2)}dx=\frac{\pi }{4}\log (1+a^2)a\ge 0$

1.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\arctan \alpha x}{x\sqrt{1-x^2}}dx=\frac{\pi }{2}\text{arsinh}\alpha$

1.4

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\arctan x}{1+2\cos \alpha x+x^2}dx=\frac{\pi }{4}\frac{\alpha }{\sin \alpha }-\frac{\pi }{2}<\text{Re}(\alpha )<\frac{\pi }{2}$

1.5

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

2.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\arctan (x)}{\exp (\pi x)-1}\mathrm{d}x=\frac{1}{2}[1-\ln (2)]$

2.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\arctan (x)}{\exp (2\pi x)-1}\mathrm{d}x=\frac{1}{4}[2-\ln (2\pi )]$

2.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\arctan (x)}{\sinh (\pi x)}\mathrm{d}x=\frac{1}{2}\ln \left(\frac{\pi }{2}\right)$

2.4

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{2\arctan (x)-x\ln (x^2+1)}{2(x^2+1)\sinh (\pi x)}\mathrm{d}x=\frac{1}{2}[2\gamma \ln (2)-\ln (2{)}^2]$