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Beliebt Rechnen >

integral of x*ln(1+x)

Lösung

\int x \cdot ln (1 + x) d x

Lösung

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

Schritte zur Lösung

\int x ln (1 + x) d x

Wende U-Substitution an

= \int (u - 1) ln (u) d u

Wende die partielle Integration an

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

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

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

Setze in

u = 1 + x

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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