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

Lösung

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

3 (\frac{9}{2} x^{2} ln^{2} (x) - 3 (\frac{3}{2} x^{2} ln (x) - \frac{3 x ^{2}}{4})) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Vereinfache

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

Wende die partielle Integration an

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

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

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

\frac{1}{2} x^{2} ln^{2} (x^{3}) = \frac{9}{2} x^{2} ln^{2} (x)

= 3 (\frac{9}{2} x^{2} ln^{2} (x) - 3 (\frac{3}{2} x^{2} ln (x) - \frac{3 x ^{2}}{4}))

Füge eine Konstante zur Lösung hinzu

= 3 (\frac{9}{2} x^{2} ln^{2} (x) - 3 (\frac{3}{2} x^{2} ln (x) - \frac{3 x ^{2}}{4})) + C

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