integral of 6x(ln(x^5))^2

Lösung

\int 6 x (ln (x^{5}))^{2} d x

6 (\frac{25}{2} x^{2} ln^{2} (x) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4})) + C

Schritte zur Lösung

\int 6 x (ln (x^{5}))^{2} d x

Entferne die Konstante:

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

= 6 \cdot \int x (ln (x^{5}))^{2} d x

Vereinfache

= 6 \cdot \int x ln^{2} (x^{5}) d x

Wende die partielle Integration an

= 6 (\frac{1}{2} x^{2} ln^{2} (x^{5}) - \int 5 x ln (x^{5}) d x)

\int 5 x ln (x^{5}) d x = 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4})

= 6 (\frac{1}{2} x^{2} ln^{2} (x^{5}) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4}))

\frac{1}{2} x^{2} ln^{2} (x^{5}) = \frac{25}{2} x^{2} ln^{2} (x)

= 6 (\frac{25}{2} x^{2} ln^{2} (x) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4}))

Füge eine Konstante zur Lösung hinzu

= 6 (\frac{25}{2} x^{2} ln^{2} (x) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4})) + C

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