integral of xln(5-x)

Lösung

\int x ln (5 - x) d x

\frac{1}{2} x^{2} ln (5 - x) + \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣) + C

Schritte zur Lösung

\int x ln (5 - x) d x

Wende die partielle Integration an

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

\int - \frac{x ^{2}}{2 ( 5 - x )} d x = - \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣)

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} ln (5 - x) + \frac{1}{2} (- \frac{x ^{2}}{2} - 5 x - 25 ln ∣ - x + 5 ∣) + C

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