integral of 2x*ln^2(x)

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{1 - x ^{2}}{2 x} d x

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