integral of xarctan(2/x)

Lösung

\int x arctan (\frac{2}{x}) d x

\frac{1}{2} x^{2} arctan (\frac{2}{x}) - 2 arctan (\frac{x}{2}) + x + C

Schritte zur Lösung

\int x arctan (\frac{2}{x}) d x

Wende die partielle Integration an

= \frac{1}{2} x^{2} arctan (\frac{2}{x}) - \int - \frac{x ^{2}}{4 + x ^{2}} d x

\int - \frac{x ^{2}}{4 + x ^{2}} d x = 2 arctan (\frac{x}{2}) - x

= \frac{1}{2} x^{2} arctan (\frac{2}{x}) - (2 arctan (\frac{x}{2}) - x)

Vereinfache

= \frac{1}{2} x^{2} arctan (\frac{2}{x}) - 2 arctan (\frac{x}{2}) + x

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} arctan (\frac{2}{x}) - 2 arctan (\frac{x}{2}) + x + C

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