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Beliebt Rechnen >

integral of ln(sqrt(x+x^2))

Lösung

\int ln (x + x^{2}) d x

Lösung

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

Schritte zur Lösung

\int ln (x + x^{2}) d x

Vereinfache

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

= \int \frac{1}{2} ln (x + x^{2}) d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int ln (x + x^{2}) d x

Wende die partielle Integration an

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

\int \frac{1 + 2 x}{1 + x} d x = 2 x - ln ∣ x + 1 ∣

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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