integral of xarctanh(x)

Lösung

\int x arctanh (x) d x

\frac{1}{2} x^{2} arctanh (x) - \frac{1}{4} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 x) + C

Schritte zur Lösung

\int x arctanh (x) d x

Wende die partielle Integration an

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

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

= \frac{1}{2} x^{2} arctanh (x) - \frac{1}{4} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} arctanh (x) - \frac{1}{4} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 x) + C

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