integral of 4/(x^2+18x+80)

Lösung

\int \frac{4}{x ^{2} + 18 x + 80} d x

4 (\frac{1}{2} ln ∣ x + 8 ∣ - \frac{1}{2} ln ∣ x + 10 ∣) + C

Schritte zur Lösung

\int \frac{4}{x ^{2} + 18 x + 80} d x

Entferne die Konstante:

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

= 4 \cdot \int \frac{1}{x ^{2} + 18 x + 80} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} + 18 x + 80} : \frac{1}{2 ( x + 8 )} - \frac{1}{2 ( x + 10 )}

= 4 \cdot \int \frac{1}{2 ( x + 8 )} - \frac{1}{2 ( x + 10 )} d x

Wende die Summenregel an:

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

= 4 (\int \frac{1}{2 ( x + 8 )} d x - \int \frac{1}{2 ( x + 10 )} d x)

\int \frac{1}{2 ( x + 8 )} d x = \frac{1}{2} ln ∣ x + 8 ∣

\int \frac{1}{2 ( x + 10 )} d x = \frac{1}{2} ln ∣ x + 10 ∣

= 4 (\frac{1}{2} ln ∣ x + 8 ∣ - \frac{1}{2} ln ∣ x + 10 ∣)

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{1}{2} ln ∣ x + 8 ∣ - \frac{1}{2} ln ∣ x + 10 ∣) + C

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