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Beliebt Rechnen >

integral of xln(x-1)

Lösung

\int x ln (x - 1) d x

Lösung

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

Schritte zur Lösung

\int x ln (x - 1) 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 = x - 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative (x^2)/(5+8x)

d er i v a t i v e \frac{x ^{2}}{5 + 8 x}

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integral of 1/(x(5+3ln(x)))

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