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Beliebt Rechnen >

integral of ((ln(x))^2)/(4x^3)

Lösung

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

Lösung

\frac{- 2 ln ^{2} ( x ) - 2 ln ( x ) - 1}{16 x ^{2}} + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

\frac{1}{4} (- \frac{ln ^{2} ( x )}{2 x ^{2}} - (\frac{ln ( x )}{2 x ^{2}} + \frac{1}{4 x ^{2}})) : \frac{- 2 ln ^{2} ( x ) - 2 ln ( x ) - 1}{16 x ^{2}}

= \frac{- 2 ln ^{2} ( x ) - 2 ln ( x ) - 1}{16 x ^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{- 2 ln ^{2} ( x ) - 2 ln ( x ) - 1}{16 x ^{2}} + C

Graph

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