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Beliebt Rechnen >

integral of ln(1+2x)

Lösung

\int ln (1 + 2 x) d x

Lösung

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

Schritte zur Lösung

\int ln (1 + 2 x) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

\int 1 d u = u

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

Setze in

u = 1 + 2 x

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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