integral of 1/(45-20x-25x^2)

Lösung

\int \frac{1}{45 - 20 x - 25 x ^{2}} d x

\frac{1}{70} (ln \frac{9}{7} + \frac{5}{7} x - ln - \frac{5}{7} + \frac{5}{7} x) + C

Schritte zur Lösung

\int \frac{1}{45 - 20 x - 25 x ^{2}} d x

Vervollständige das Quadrat

45 - 20 x - 25 x^{2} : - 25 (x + \frac{2}{5})^{2} + 49

= \int \frac{1}{- 25 ( x + \frac{2}{5} ) ^{2} + 49} d x

Wende U-Substitution an

= \int \frac{1}{- 25 u ^{2} + 49} d u

Wende integrale Substitution an

= \int \frac{1}{35 ( - v ^{2} + 1 )} d v

Entferne die Konstante:

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

= \frac{1}{35} \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}{35} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Ersetze zurück

= \frac{1}{35} (\frac{ln \frac{5}{7} ( x + \frac{2}{5} ) + 1}{2} - \frac{ln \frac{5}{7} ( x + \frac{2}{5} ) - 1}{2})

Vereinfache

\frac{1}{35} (\frac{ln \frac{5}{7} ( x + \frac{2}{5} ) + 1}{2} - \frac{ln \frac{5}{7} ( x + \frac{2}{5} ) - 1}{2}) : \frac{1}{70} (ln \frac{9}{7} + \frac{5}{7} x - ln - \frac{5}{7} + \frac{5}{7} x)

= \frac{1}{70} (ln \frac{9}{7} + \frac{5}{7} x - ln - \frac{5}{7} + \frac{5}{7} x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{70} (ln \frac{9}{7} + \frac{5}{7} x - ln - \frac{5}{7} + \frac{5}{7} x) + C

\int \frac{1}{x ^{4} + 13 x ^{2} + 36} d x

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