integral of x/(sqrt(x^4-2x^2+10))

Lösung

\int \frac{x}{x ^{4} - 2 x ^{2} + 10} d x

\frac{1}{2} ln (\frac{x ^{2} - 1 + x ^{4} - 2 x ^{2} + 10}{3}) + C

Schritte zur Lösung

\int \frac{x}{x ^{4} - 2 x ^{2} + 10} d x

Wende U-Substitution an

= \int \frac{1}{2 u ^{2} - 2 u + 10} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{2} - 2 u + 10} d u

Vervollständige das Quadrat

u^{2} - 2 u + 10 : (u - 1)^{2} + 9

= \frac{1}{2} \cdot \int \frac{1}{( u - 1 ) ^{2} + 9} d u

Wende U-Substitution an

= \frac{1}{2} \cdot \int \frac{1}{v ^{2} + 9} d v

Trigonometrische Substitution anwenden

= \frac{1}{2} \cdot \int sec (w) d w

Nutze das gemeinsame Integral :

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

= \frac{1}{2} ln ∣ tan (w) + sec (w) ∣

Ersetze zurück

= \frac{1}{2} ln tan (arctan (\frac{1}{3} (x^{2} - 1))) + sec (arctan (\frac{1}{3} (x^{2} - 1)))

Vereinfache

\frac{1}{2} ln tan (arctan (\frac{1}{3} (x^{2} - 1))) + sec (arctan (\frac{1}{3} (x^{2} - 1))) : \frac{1}{2} ln (\frac{x ^{2} - 1 + x ^{4} - 2 x ^{2} + 10}{3})

= \frac{1}{2} ln (\frac{x ^{2} - 1 + x ^{4} - 2 x ^{2} + 10}{3})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln (\frac{x ^{2} - 1 + x ^{4} - 2 x ^{2} + 10}{3}) + C

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