integral of ln(x^2-9)

Lösung

\int ln (x^{2} - 9) d x

x ln (x^{2} - 9) + 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) + C

Schritte zur Lösung

\int ln (x^{2} - 9) d x

Wende die partielle Integration an

= x ln (x^{2} - 9) - \int \frac{2 x ^{2}}{x ^{2} - 9} d x

\int \frac{2 x ^{2}}{x ^{2} - 9} d x = - 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3})

= x ln (x^{2} - 9) - (- 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}))

Vereinfache

= x ln (x^{2} - 9) + 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3})

Füge eine Konstante zur Lösung hinzu

= x ln (x^{2} - 9) + 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) + C

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