integral from 0 to 1/2 of (arctan(x))/x

Lösung

\int_{0}^{\frac{1}{2}} \frac{arctan ( x )}{x} d x

\frac{i ( Li _{2} ( - i \frac{1}{2} ) - Li _{2} ( i \frac{1}{2} ) ) - i ( Li _{2} ( - 0 i ) - Li _{2} ( 0 i ) )}{2}

Schritte zur Lösung

\int_{0}^{\frac{1}{2}} \frac{arctan ( x )}{x} d x

Das ist ein nicht elementares Integral :

\int \frac{arctan ( x )}{x} d x = \frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x))

= [\frac{1}{2} i (Li_{2} (- i x) - Li_{2} (i x))]_{0}^{\frac{1}{2}}

Berechne die Grenzen:

\frac{i ( Li _{2} ( - i \frac{1}{2} ) - Li _{2} ( i \frac{1}{2} ) ) - i ( Li _{2} ( - 0 i ) - Li _{2} ( 0 i ) )}{2}

= \frac{i ( Li _{2} ( - i \frac{1}{2} ) - Li _{2} ( i \frac{1}{2} ) ) - i ( Li _{2} ( - 0 i ) - Li _{2} ( 0 i ) )}{2}

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