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Beliebt Rechnen >

integral of (3x+5)/(x^3-x^2-x+1)

Lösung

\int \frac{3 x + 5}{x ^{3} - x ^{2} - x + 1} d x

Lösung

- \frac{1}{2} ln ∣ x - 1 ∣ - \frac{4}{x - 1} + \frac{1}{2} ln ∣ x + 1 ∣ + C

Schritte zur Lösung

\int \frac{3 x + 5}{x ^{3} - x ^{2} - x + 1} d x

Faktorisiere

\frac{3 x + 5}{x ^{3} - x ^{2} - x + 1}

= \int \frac{3 x + 5}{( x - 1 ) ^{2} ( x + 1 )} d x

Ermittle den Partialbruch von

\frac{3 x + 5}{( x - 1 ) ^{2} ( x + 1 )} : - \frac{1}{2 ( x - 1 )} + \frac{4}{( x - 1 ) ^{2}} + \frac{1}{2 ( x + 1 )}

= \int - \frac{1}{2 ( x - 1 )} + \frac{4}{( x - 1 ) ^{2}} + \frac{1}{2 ( x + 1 )} 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}{2 ( x - 1 )} d x + \int \frac{4}{( x - 1 ) ^{2}} d x + \int \frac{1}{2 ( x + 1 )} d x

\int \frac{1}{2 ( x - 1 )} d x = \frac{1}{2} ln ∣ x - 1 ∣

\int \frac{4}{( x - 1 ) ^{2}} d x = - \frac{4}{x - 1}

\int \frac{1}{2 ( x + 1 )} d x = \frac{1}{2} ln ∣ x + 1 ∣

= - \frac{1}{2} ln ∣ x - 1 ∣ - \frac{4}{x - 1} + \frac{1}{2} ln ∣ x + 1 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln ∣ x - 1 ∣ - \frac{4}{x - 1} + \frac{1}{2} ln ∣ x + 1 ∣ + C

Graph

Beliebte Beispiele

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