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Beliebt Rechnen >

integral of xln(2+x^2)

Lösung

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

Lösung

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

Schritte zur Lösung

\int x ln (2 + 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 = 2 + x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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