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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x ∣ + \frac{2}{x + 1} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - (- \frac{2}{x + 1})

Vereinfache

= ln ∣ x ∣ + \frac{2}{x + 1}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + \frac{2}{x + 1} + C

Graph

Beliebte Beispiele

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