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Beliebt Rechnen >

integral of 12x^3(ln(4x))^2

Lösung

\int 12 x^{3} (ln (4 x))^{2} d x

Lösung

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

Schritte zur Lösung

\int 12 x^{3} (ln (4 x))^{2} d x

Entferne die Konstante:

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

= 12 \cdot \int x^{3} (ln (4 x))^{2} d x

Vereinfache

= 12 \cdot \int x^{3} ln^{2} (4 x) d x

Wende die partielle Integration an

= 12 (\frac{1}{4} x^{4} ln^{2} (4 x) - \int \frac{1}{2} x^{3} ln (4 x) d x)

\int \frac{1}{2} x^{3} ln (4 x) d x = \frac{1}{2} (\frac{1}{4} x^{4} ln (4 x) - \frac{x ^{4}}{16})

= 12 (\frac{1}{4} x^{4} ln^{2} (4 x) - \frac{1}{2} (\frac{1}{4} x^{4} ln (4 x) - \frac{x ^{4}}{16}))

Füge eine Konstante zur Lösung hinzu

= 12 (\frac{1}{4} x^{4} ln^{2} (4 x) - \frac{1}{2} (\frac{1}{4} x^{4} ln (4 x) - \frac{x ^{4}}{16})) + C

Graph

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