integral of 2/(sqrt(x^2-4x+9))

Lösung

\int \frac{2}{x ^{2} - 4 x + 9} d x

2 ln \frac{1}{5} (x - 2) + \frac{1}{5} (x^{2} - 4 x + 9) + C

Schritte zur Lösung

\int \frac{2}{x ^{2} - 4 x + 9} d x

Entferne die Konstante:

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

= 2 \cdot \int \frac{1}{x ^{2} - 4 x + 9} d x

Vervollständige das Quadrat

x^{2} - 4 x + 9 : (x - 2)^{2} + 5

= 2 \cdot \int \frac{1}{( x - 2 ) ^{2} + 5} d x

Wende U-Substitution an

= 2 \cdot \int \frac{1}{u ^{2} + 5} d u

Trigonometrische Substitution anwenden

= 2 \cdot \int sec (v) d v

Nutze das gemeinsame Integral :

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= 2 ln ∣ tan (v) + sec (v) ∣

Ersetze zurück

= 2 ln tan (arctan (\frac{1}{5} (x - 2))) + sec (arctan (\frac{1}{5} (x - 2)))

Vereinfache

2 ln tan (arctan (\frac{1}{5} (x - 2))) + sec (arctan (\frac{1}{5} (x - 2))) : 2 ln \frac{1}{5} (x - 2) + \frac{1}{5} (x^{2} - 4 x + 9)

= 2 ln \frac{1}{5} (x - 2) + \frac{1}{5} (x^{2} - 4 x + 9)

Füge eine Konstante zur Lösung hinzu

= 2 ln \frac{1}{5} (x - 2) + \frac{1}{5} (x^{2} - 4 x + 9) + C

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