integral of x*(arctan(x))^2

Lösung

\int x \cdot (arctan (x))^{2} d x

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

Schritte zur Lösung

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

Vereinfache

= \int x arctan^{2} (x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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