integral of (xln(x^2+3))/(x^2+3)

Lösung

\int \frac{x ln ( x ^{2} + 3 )}{x ^{2} + 3} d x

\frac{1}{4} ln^{2} (x^{2} + 3) + C

Schritte zur Lösung

\int \frac{x ln ( x ^{2} + 3 )}{x ^{2} + 3} d x

Wende U-Substitution an

= \int \frac{ln ( u + 3 )}{2 ( u + 3 )} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{ln ( u + 3 )}{u + 3} d u

Wende U-Substitution an

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

Wende U-Substitution an

= \frac{1}{2} \cdot \int w d w

Wende die Potenzregel an

= \frac{1}{2} \cdot \frac{w ^{2}}{2}

Ersetze zurück

= \frac{1}{2} \cdot \frac{ln ^{2} ( x ^{2} + 3 )}{2}

Vereinfache

\frac{1}{2} \cdot \frac{ln ^{2} ( x ^{2} + 3 )}{2} : \frac{1}{4} ln^{2} (x^{2} + 3)

= \frac{1}{4} ln^{2} (x^{2} + 3)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} ln^{2} (x^{2} + 3) + C

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