integral of ln(x)(sqrt(1-(ln(x))^2))/x

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int u 1 - u^{2} d u

Wende U-Substitution an

= \int - v^{2} d v

Entferne die Konstante:

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

= - \int v^{2} d v

Wende die Potenzregel an

= - \frac{v ^{3}}{3}

Ersetze zurück

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

Vereinfache

- \frac{( 1 - ln ^{2} ( x ) ) ^{3}}{3} : - \frac{1}{3} (1 - ln^{2} (x))^{\frac{3}{2}}

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

Füge eine Konstante zur Lösung hinzu

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

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