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Beliebt Rechnen >

integral of (3x^2-8x+13)/((x+3)(x-1)^2)

Lösung

\int \frac{3 x ^{2} - 8 x + 13}{( x + 3 ) ( x - 1 ) ^{2}} d x

Lösung

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

Schritte zur Lösung

\int \frac{3 x ^{2} - 8 x + 13}{( x + 3 ) ( x - 1 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{3 x ^{2} - 8 x + 13}{( x + 3 ) ( x - 1 ) ^{2}} : \frac{4}{x + 3} - \frac{1}{x - 1} + \frac{2}{( x - 1 ) ^{2}}

= \int \frac{4}{x + 3} - \frac{1}{x - 1} + \frac{2}{( x - 1 ) ^{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{4}{x + 3} d x - \int \frac{1}{x - 1} d x + \int \frac{2}{( x - 1 ) ^{2}} d x

\int \frac{4}{x + 3} d x = 4 ln ∣ x + 3 ∣

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

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

= 4 ln ∣ x + 3 ∣ - ln ∣ x - 1 ∣ - \frac{2}{x - 1}

Füge eine Konstante zur Lösung hinzu

= 4 ln ∣ x + 3 ∣ - ln ∣ x - 1 ∣ - \frac{2}{x - 1} + C

Graph

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