integral of-x*ln(1+x^2)

Lösung

\int - x \cdot ln (1 + x^{2}) d x

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

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Wende U-Substitution an

= - \int \frac{ln ( u )}{2} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{2} \cdot \int ln (u) d u

Wende die partielle Integration an

= - \frac{1}{2} (u ln (u) - \int 1 d u)

\int 1 d u = u

= - \frac{1}{2} (u ln (u) - u)

Setze in

u = 1 + x^{2}

ein

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

Multipliziere aus

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

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

Füge eine Konstante zur Lösung hinzu

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

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