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Beliebt Rechnen >

integral of 5/(x^2+7x+12)

Lösung

\int \frac{5}{x ^{2} + 7 x + 12} d x

Lösung

5 (ln ∣ x + 3 ∣ - ln ∣ x + 4 ∣) + C

Schritte zur Lösung

\int \frac{5}{x ^{2} + 7 x + 12} d x

Entferne die Konstante:

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

= 5 \cdot \int \frac{1}{x ^{2} + 7 x + 12} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} + 7 x + 12} : \frac{1}{x + 3} - \frac{1}{x + 4}

= 5 \cdot \int \frac{1}{x + 3} - \frac{1}{x + 4} 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

= 5 (\int \frac{1}{x + 3} d x - \int \frac{1}{x + 4} d x)

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

\int \frac{1}{x + 4} d x = ln ∣ x + 4 ∣

= 5 (ln ∣ x + 3 ∣ - ln ∣ x + 4 ∣)

Füge eine Konstante zur Lösung hinzu

= 5 (ln ∣ x + 3 ∣ - ln ∣ x + 4 ∣) + C

Graph

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