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Beliebt Rechnen >

integral from-3 to 3 of (10)/(x^2-2x-24)

Lösung

\int_{- 3}^{3} \frac{10}{x ^{2} - 2 x - 24} d x

Lösung

- ln (7) - ln (3)

+1

Dezimale

- 3.04452 \dots

Schritte zur Lösung

\int_{- 3}^{3} \frac{10}{x ^{2} - 2 x - 24} d x

Entferne die Konstante:

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

= 10 \cdot \int_{- 3}^{3} \frac{1}{x ^{2} - 2 x - 24} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} - 2 x - 24} : - \frac{1}{10 ( x + 4 )} + \frac{1}{10 ( x - 6 )}

= 10 \cdot \int_{- 3}^{3} - \frac{1}{10 ( x + 4 )} + \frac{1}{10 ( x - 6 )} 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_{- 3}^{3} \frac{1}{10 ( x + 4 )} d x + \int_{- 3}^{3} \frac{1}{10 ( x - 6 )} d x)

\int_{- 3}^{3} \frac{1}{10 ( x + 4 )} d x = \frac{1}{10} ln (7)

\int_{- 3}^{3} \frac{1}{10 ( x - 6 )} d x = - \frac{1}{10} ln (3)

= 10 (- \frac{1}{10} ln (7) - \frac{1}{10} ln (3))

Vereinfache

10 (- \frac{1}{10} ln (7) - \frac{1}{10} ln (3)) : - ln (7) - ln (3)

= - ln (7) - ln (3)

Graph

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