integral from 2 to 3 of (sqrt(x^2-4))/x

Lösung

\int_{2}^{3} \frac{x ^{2} - 4}{x} d x

- 2 arcsec (\frac{3}{2}) + 5

Dezimale

0.55393 \dots

Schritte zur Lösung

\int_{2}^{3} \frac{x ^{2} - 4}{x} d x

Trigonometrische Substitution anwenden

= \int_{0}^{arcsec (\frac{3}{2})} 2 tan^{2} (u) d u

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{arcsec (\frac{3}{2})} tan^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 2 \cdot \int_{0}^{arcsec (\frac{3}{2})} - 1 + sec^{2} (u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (- \int_{0}^{arcsec (\frac{3}{2})} 1 d u + \int_{0}^{arcsec (\frac{3}{2})} sec^{2} (u) d u)

\int_{0}^{arcsec (\frac{3}{2})} 1 d u = arcsec (\frac{3}{2})

\int_{0}^{arcsec (\frac{3}{2})} sec^{2} (u) d u = \frac{5}{2}

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

Vereinfache

2 (- arcsec (\frac{3}{2}) + \frac{5}{2}) : - 2 arcsec (\frac{3}{2}) + 5

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

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