integral of-xln(x^2+1)

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

ein

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

Multipliziere aus

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

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

Füge eine Konstante zur Lösung hinzu

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