integral of sqrt(x)*ln^2(x)

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int \frac{4}{3} x^{\frac{1}{2}} ln (x) d x = \frac{4}{3} (\frac{2}{3} x^{\frac{3}{2}} ln (x) - \frac{4}{9} x^{\frac{3}{2}})

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

Füge eine Konstante zur Lösung hinzu

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

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