integral of 1/(sqrt(x^2-10x+29))

Lösung

\int \frac{1}{x ^{2} - 10 x + 29} d x

ln (\frac{1}{2} x - 5 + x^{2} - 10 x + 29) + C

Schritte zur Lösung

\int \frac{1}{x ^{2} - 10 x + 29} d x

Vervollständige das Quadrat

x^{2} - 10 x + 29 : (x - 5)^{2} + 4

= \int \frac{1}{( x - 5 ) ^{2} + 4} d x

Wende U-Substitution an

= \int \frac{1}{u ^{2} + 4} d u

Trigonometrische Substitution anwenden

= \int sec (v) d v

Nutze das gemeinsame Integral :

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

= ln ∣ tan (v) + sec (v) ∣

Ersetze zurück

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

Vereinfache

ln tan (arctan (\frac{1}{2} (x - 5))) + sec (arctan (\frac{1}{2} (x - 5))) : ln (\frac{1}{2} x - 5 + x^{2} - 10 x + 29)

= ln (\frac{1}{2} x - 5 + x^{2} - 10 x + 29)

Füge eine Konstante zur Lösung hinzu

= ln (\frac{1}{2} x - 5 + x^{2} - 10 x + 29) + C

\int \frac{4 x ^{2} + 3 x + 3}{x ^{3} + x ^{2} + x + 1} d x

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