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Beliebt Rechnen >

integral of 8(x+1)ln(x+1)

Lösung

\int 8 (x + 1) ln (x + 1) d x

Lösung

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

Schritte zur Lösung

\int 8 (x + 1) ln (x + 1) d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 8 \cdot \int (x + 1) ln (x + 1) d x

Wende U-Substitution an

= 8 \cdot \int u ln (u) d u

Wende die partielle Integration an

= 8 (\frac{1}{2} u^{2} ln (u) - \int \frac{u}{2} d u)

\int \frac{u}{2} d u = \frac{u ^{2}}{4}

= 8 (\frac{1}{2} u^{2} ln (u) - \frac{u ^{2}}{4})

Setze in

u = x + 1

ein

= 8 (\frac{1}{2} (x + 1)^{2} ln (x + 1) - \frac{( x + 1 ) ^{2}}{4})

Vereinfache

8 (\frac{1}{2} (x + 1)^{2} ln (x + 1) - \frac{( x + 1 ) ^{2}}{4}) : 8 (\frac{1}{2} x^{2} ln (x + 1) + x ln (x + 1) + \frac{1}{2} ln (x + 1) - \frac{1}{4} (x + 1)^{2})

= 8 (\frac{1}{2} x^{2} ln (x + 1) + x ln (x + 1) + \frac{1}{2} ln (x + 1) - \frac{1}{4} (x + 1)^{2})

Füge eine Konstante zur Lösung hinzu

= 8 (\frac{1}{2} x^{2} ln (x + 1) + x ln (x + 1) + \frac{1}{2} ln (x + 1) - \frac{1}{4} (x + 1)^{2}) + C

Graph

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