integral of (x^3+1)ln(x)

Lösung

\int (x^{3} + 1) ln (x) d x

\frac{1}{4} x^{4} ln (x) + x ln (x) - \frac{x ^{4}}{16} - x + C

Schritte zur Lösung

\int (x^{3} + 1) ln (x) d x

Wende die partielle Integration an

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

\int \frac{1}{4} (x^{3} + 4) d x = \frac{x ^{4}}{16} + x

= ln (x) (\frac{x ^{4}}{4} + x) - (\frac{x ^{4}}{16} + x)

Vereinfache

ln (x) (\frac{x ^{4}}{4} + x) - (\frac{x ^{4}}{16} + x) : \frac{1}{4} x^{4} ln (x) + x ln (x) - \frac{x ^{4}}{16} - x

= \frac{1}{4} x^{4} ln (x) + x ln (x) - \frac{x ^{4}}{16} - x

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} x^{4} ln (x) + x ln (x) - \frac{x ^{4}}{16} - x + C