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Beliebt Rechnen >

integral of (144)/(x^3-12x^2)

Lösung

\int \frac{144}{x ^{3} - 12 x ^{2}} d x

Lösung

- ln ∣ x ∣ + \frac{12}{x} + ln ∣ x - 12 ∣ + C

Schritte zur Lösung

\int \frac{144}{x ^{3} - 12 x ^{2}} d x

Entferne die Konstante:

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

= 144 \cdot \int \frac{1}{x ^{3} - 12 x ^{2}} d x

Ermittle den Partialbruch von

\frac{1}{x ^{3} - 12 x ^{2}} : - \frac{1}{144 x} - \frac{1}{12 x ^{2}} + \frac{1}{144 ( x - 12 )}

= 144 \cdot \int - \frac{1}{144 x} - \frac{1}{12 x ^{2}} + \frac{1}{144 ( x - 12 )} 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

= 144 (- \int \frac{1}{144 x} d x - \int \frac{1}{12 x ^{2}} d x + \int \frac{1}{144 ( x - 12 )} d x)

\int \frac{1}{144 x} d x = \frac{1}{144} ln ∣ x ∣

\int \frac{1}{12 x ^{2}} d x = - \frac{1}{12 x}

\int \frac{1}{144 ( x - 12 )} d x = \frac{1}{144} ln ∣ x - 12 ∣

= 144 (- \frac{1}{144} ln ∣ x ∣ - (- \frac{1}{12 x}) + \frac{1}{144} ln ∣ x - 12 ∣)

Vereinfache

144 (- \frac{1}{144} ln ∣ x ∣ - (- \frac{1}{12 x}) + \frac{1}{144} ln ∣ x - 12 ∣) : - ln ∣ x ∣ + \frac{12}{x} + ln ∣ x - 12 ∣

= - ln ∣ x ∣ + \frac{12}{x} + ln ∣ x - 12 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x ∣ + \frac{12}{x} + ln ∣ x - 12 ∣ + C

Graph

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