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Beliebt Rechnen >

integral of ((2ln(x))^2)/(4x)

Lösung

\int \frac{( 2 ln ( x ) ) ^{2}}{4 x} d x

Lösung

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

Schritte zur Lösung

\int \frac{( 2 ln ( x ) ) ^{2}}{4 x} d x

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int \frac{( 2 ln ( x ) ) ^{2}}{x} d x

Wende die partielle Integration an

= \frac{1}{4} (4 ln^{2} (x) - \int \frac{8 ln ( x ) - 4 ln ^{2} ( x )}{x} d x)

\int \frac{8 ln ( x ) - 4 ln ^{2} ( x )}{x} d x = 4 ln^{2} (x) - \frac{4}{3} ln^{3} (x)

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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