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Beliebt Rechnen >

integral of (10)/((x-1)(x^2-9))

Lösung

\int \frac{10}{( x - 1 ) ( x ^{2} - 9 )} d x

Lösung

10 (- \frac{1}{8} ln ∣ x - 1 ∣ + \frac{1}{24} ln ∣ x + 3 ∣ + \frac{1}{12} ln ∣ x - 3 ∣) + C

Schritte zur Lösung

\int \frac{10}{( x - 1 ) ( x ^{2} - 9 )} d x

Entferne die Konstante:

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

= 10 \cdot \int \frac{1}{( x - 1 ) ( x ^{2} - 9 )} d x

Ermittle den Partialbruch von

\frac{1}{( x - 1 ) ( x ^{2} - 9 )} : - \frac{1}{8 ( x - 1 )} + \frac{1}{24 ( x + 3 )} + \frac{1}{12 ( x - 3 )}

= 10 \cdot \int - \frac{1}{8 ( x - 1 )} + \frac{1}{24 ( x + 3 )} + \frac{1}{12 ( x - 3 )} 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

= 10 (- \int \frac{1}{8 ( x - 1 )} d x + \int \frac{1}{24 ( x + 3 )} d x + \int \frac{1}{12 ( x - 3 )} d x)

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

\int \frac{1}{24 ( x + 3 )} d x = \frac{1}{24} ln ∣ x + 3 ∣

\int \frac{1}{12 ( x - 3 )} d x = \frac{1}{12} ln ∣ x - 3 ∣

= 10 (- \frac{1}{8} ln ∣ x - 1 ∣ + \frac{1}{24} ln ∣ x + 3 ∣ + \frac{1}{12} ln ∣ x - 3 ∣)

Füge eine Konstante zur Lösung hinzu

= 10 (- \frac{1}{8} ln ∣ x - 1 ∣ + \frac{1}{24} ln ∣ x + 3 ∣ + \frac{1}{12} ln ∣ x - 3 ∣) + C

Graph

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