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

Lösung

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

- \frac{2}{3} ln ∣ x + 1 ∣ + \frac{5}{3} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{2}{3 ( x + 1 )} d x = \frac{2}{3} ln ∣ x + 1 ∣

\int \frac{5}{3 ( x - 2 )} d x = \frac{5}{3} ln ∣ x - 2 ∣

= - \frac{2}{3} ln ∣ x + 1 ∣ + \frac{5}{3} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{3} ln ∣ x + 1 ∣ + \frac{5}{3} ln ∣ x - 2 ∣ + C

\int \frac{1}{9 x + 4} d x

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