integral of sqrt(x)ln(sqrt(x))

Lösung

\int x ln (x) d x

\frac{1}{3} x^{\frac{3}{2}} ln (x) - \frac{2}{9} x^{\frac{3}{2}} + C

Schritte zur Lösung

\int x ln (x) d x

Wende die partielle Integration an

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

\int \frac{1}{3} x^{\frac{1}{2}} d x = \frac{2}{9} x^{\frac{3}{2}}

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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