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Beliebt Rechnen >

integral of (3x^2+8x-12)/(x^3+7x^2+12x)

Lösung

\int \frac{3 x ^{2} + 8 x - 12}{x ^{3} + 7 x ^{2} + 12 x} d x

Lösung

- ln ∣ x ∣ + 3 ln ∣ x + 3 ∣ + ln ∣ x + 4 ∣ + C

Schritte zur Lösung

\int \frac{3 x ^{2} + 8 x - 12}{x ^{3} + 7 x ^{2} + 12 x} d x

Ermittle den Partialbruch von

\frac{3 x ^{2} + 8 x - 12}{x ^{3} + 7 x ^{2} + 12 x} : - \frac{1}{x} + \frac{3}{x + 3} + \frac{1}{x + 4}

= \int - \frac{1}{x} + \frac{3}{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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

= - ln ∣ x ∣ + 3 ln ∣ x + 3 ∣ + ln ∣ x + 4 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x ∣ + 3 ln ∣ x + 3 ∣ + ln ∣ x + 4 ∣ + C

Graph

Beliebte Beispiele

integral of x^3sqrt(x^4+8)

\int x^{3} x^{4} + 8 d x

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\int x e^{\frac{x ^{2}}{5}} d x

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\int \frac{2 t ^{3} - 4 t ^{2} + 1}{2 t} d t

integral of x^3sin(4x^4-pi)

\int x^{3} sin (4 x^{4} - π) d x

integral of (3x^2-3x+1)/(x^3+x)

\int \frac{3 x ^{2} - 3 x + 1}{x ^{3} + x} d x