integral from 2 to 5 of ln(x^2-2)

Lösung

\int_{2}^{5} ln (x^{2} - 2) d x

5 ln (23) + 2 ln (\frac{5}{2} + 1) + 2 ln (2 - 1) - 6 - 2 ln (2) - 2 ln (\frac{5}{2} - 1) - 2 ln (1 + 2)

Dezimale

6.62069 \dots

Schritte zur Lösung

\int_{2}^{5} ln (x^{2} - 2) d x

Wende die partielle Integration an

= [x ln (x^{2} - 2) - \int \frac{2 x ^{2}}{x ^{2} - 2} d x]_{2}^{5}

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

= [x ln (x^{2} - 2) - (- 2 ln \frac{x}{2} + 1 + 2 ln \frac{x}{2} - 1 + 2 x)]_{2}^{5}

Vereinfache

= [x ln (x^{2} - 2) + 2 ln \frac{x}{2} + 1 - 2 ln \frac{x}{2} - 1 - 2 x]_{2}^{5}

Berechne die Grenzen:

5 ln (23) + 2 ln (\frac{5}{2} + 1) + 2 ln (2 - 1) - 6 - 2 ln (2) - 2 ln (\frac{5}{2} - 1) - 2 ln (1 + 2)

= 5 ln (23) + 2 ln (\frac{5}{2} + 1) + 2 ln (2 - 1) - 6 - 2 ln (2) - 2 ln (\frac{5}{2} - 1) - 2 ln (1 + 2)

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