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integral of (x^2-2x+1)/(sqrt(x^2-2x+17))

Lösung

\int \frac{x ^{2} - 2 x + 1}{x ^{2} - 2 x + 17} d x

16 (- ln (\frac{1}{4} x - 1 + x^{2} - 2 x + 17) + \frac{1}{32} (x - 1) x^{2} - 2 x + 17 + \frac{1}{2} ln (\frac{x - 1 + x ^{2} - 2 x + 17}{4})) + C

Schritte zur Lösung

\int \frac{x ^{2} - 2 x + 1}{x ^{2} - 2 x + 17} d x

Vervollständige das Quadrat

x^{2} - 2 x + 1 : (x - 1)^{2}

= \int \frac{( x - 1 ) ^{2}}{x ^{2} - 2 x + 17} d x

Wende U-Substitution an

= \int \frac{u ^{2}}{u ^{2} + 16} d u

Trigonometrische Substitution anwenden

= \int 16 tan^{2} (v) sec (v) d v

Entferne die Konstante:

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

= 16 \cdot \int tan^{2} (v) sec (v) d v

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 16 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

Multipliziere aus

(- 1 + sec^{2} (v)) sec (v) : - sec (v) + sec^{3} (v)

= 16 \cdot \int - sec (v) + sec^{3} (v) d v

Wende die Summenregel an:

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

= 16 (- \int sec (v) d v + \int sec^{3} (v) d v)

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

\int sec^{3} (v) d v = \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

= 16 (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Ersetze zurück

= 16 (- ln tan (arctan (\frac{1}{4} (x - 1))) + sec (arctan (\frac{1}{4} (x - 1))) + \frac{1}{2} sec (arctan (\frac{1}{4} (x - 1))) tan (arctan (\frac{1}{4} (x - 1))) + \frac{1}{2} ln tan (arctan (\frac{1}{4} (x - 1))) + sec (arctan (\frac{1}{4} (x - 1))))

Vereinfache

16 (- ln tan (arctan (\frac{1}{4} (x - 1))) + sec (arctan (\frac{1}{4} (x - 1))) + \frac{1}{2} sec (arctan (\frac{1}{4} (x - 1))) tan (arctan (\frac{1}{4} (x - 1))) + \frac{1}{2} ln tan (arctan (\frac{1}{4} (x - 1))) + sec (arctan (\frac{1}{4} (x - 1)))) : 16 (- ln (\frac{1}{4} x - 1 + x^{2} - 2 x + 17) + \frac{1}{32} (x - 1) x^{2} - 2 x + 17 + \frac{1}{2} ln (\frac{x - 1 + x ^{2} - 2 x + 17}{4}))

= 16 (- ln (\frac{1}{4} x - 1 + x^{2} - 2 x + 17) + \frac{1}{32} (x - 1) x^{2} - 2 x + 17 + \frac{1}{2} ln (\frac{x - 1 + x ^{2} - 2 x + 17}{4}))

Füge eine Konstante zur Lösung hinzu

= 16 (- ln (\frac{1}{4} x - 1 + x^{2} - 2 x + 17) + \frac{1}{32} (x - 1) x^{2} - 2 x + 17 + \frac{1}{2} ln (\frac{x - 1 + x ^{2} - 2 x + 17}{4})) + C

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