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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

= 2 \cdot \int x (ln (x))^{3} d x

Vereinfache

= 2 \cdot \int x ln^{3} (x) d x

Wende die partielle Integration an

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

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

= 2 (\frac{1}{2} x^{2} ln^{3} (x) - \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2}))

Vereinfache

2 (\frac{1}{2} x^{2} ln^{3} (x) - \frac{3}{8} (2 x^{2} ln^{2} (x) - 2 x^{2} ln (x) + x^{2})) : x^{2} ln^{3} (x) - \frac{3}{2} x^{2} ln^{2} (x) + \frac{3}{2} x^{2} ln (x) - \frac{3}{4} x^{2}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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