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

Lösung

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

- \frac{1}{2 11} (ln \frac{x + 1}{11} + 1 - ln \frac{x + 1}{11} - 1) + C

Schritte zur Lösung

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

Faktorisiere

x^{2} + 2 x - 10 : - (- x^{2} - 2 x + 10)

= \int \frac{1}{- ( - x ^{2} - 2 x + 10 )} d x

Entferne die Konstante:

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

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

Vervollständige das Quadrat

- x^{2} - 2 x + 10 : - (x + 1)^{2} + 11

= - \int \frac{1}{- ( x + 1 ) ^{2} + 11} d x

Wende U-Substitution an

= - \int \frac{1}{- u ^{2} + 11} d u

Wende integrale Substitution an

= - \int \frac{1}{11 ( - v ^{2} + 1 )} d v

Entferne die Konstante:

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

= - \frac{1}{11} \cdot \int \frac{1}{- v ^{2} + 1} d v

Nutze das gemeinsame Integral :

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= - \frac{1}{11} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Ersetze zurück

= - \frac{1}{11} \frac{ln \frac{x + 1}{11} + 1}{2} - \frac{ln \frac{x + 1}{11} - 1}{2}

Vereinfache

- \frac{1}{11} \frac{ln \frac{x + 1}{11} + 1}{2} - \frac{ln \frac{x + 1}{11} - 1}{2} : - \frac{1}{2 11} (ln \frac{x + 1}{11} + 1 - ln \frac{x + 1}{11} - 1)

= - \frac{1}{2 11} (ln \frac{x + 1}{11} + 1 - ln \frac{x + 1}{11} - 1)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2 11} (ln \frac{x + 1}{11} + 1 - ln \frac{x + 1}{11} - 1) + C

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