integral from 0 to 2 of ln(9-x^2)

Lösung

\int_{0}^{2} ln (9 - x^{2}) d x

5 ln (5) - 4

Dezimale

4.04718 \dots

Schritte zur Lösung

\int_{0}^{2} ln (9 - x^{2}) d x

Wende die partielle Integration an

= [x ln (9 - x^{2}) - \int - \frac{2 x ^{2}}{9 - x ^{2}} d x]_{0}^{2}

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

= [x ln (9 - x^{2}) - (- 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}))]_{0}^{2}

Vereinfache

= [x ln (9 - x^{2}) + 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3})]_{0}^{2}

Berechne die Grenzen:

2 ln (5) + 6 (\frac{ln ( 3 ) + ln ( \frac{5}{3} )}{2} - \frac{2}{3})

= 2 ln (5) + 6 (\frac{ln ( 3 ) + ln ( \frac{5}{3} )}{2} - \frac{2}{3})

Vereinfache

= 5 ln (5) - 4

\int_{1}^{2} 4 x d x

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