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Beliebt Rechnen >

integral of ln(sqrt(x^2-4))

Lösung

\int ln (x^{2} - 4) d x

Lösung

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

Schritte zur Lösung

\int ln (x^{2} - 4) d x

Vereinfache

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

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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