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Beliebt Rechnen >

integral of ln(2t+1)

Lösung

\int ln (2 t + 1) d t

Lösung

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

Schritte zur Lösung

\int ln (2 t + 1) d t

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 = 2 t + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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