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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{4}{11} ln \frac{x ^{2}}{2} + 1 + \frac{9}{11 2} arctan (\frac{x}{2}) - \frac{8}{11} ln ∣ x + 3 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{3 x + 1}{( x ^{2} + 2 ) ( x + 3 )} : \frac{8 x + 9}{11 ( x ^{2} + 2 )} - \frac{8}{11 ( x + 3 )}

= \int \frac{8 x + 9}{11 ( x ^{2} + 2 )} - \frac{8}{11 ( x + 3 )} 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{8 x + 9}{11 ( x ^{2} + 2 )} d x - \int \frac{8}{11 ( x + 3 )} d x

\int \frac{8 x + 9}{11 ( x ^{2} + 2 )} d x = \frac{4}{11} ln \frac{x ^{2}}{2} + 1 + \frac{9}{11 2} arctan (\frac{x}{2})

\int \frac{8}{11 ( x + 3 )} d x = \frac{8}{11} ln ∣ x + 3 ∣

= \frac{4}{11} ln \frac{x ^{2}}{2} + 1 + \frac{9}{11 2} arctan (\frac{x}{2}) - \frac{8}{11} ln ∣ x + 3 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{4}{11} ln \frac{x ^{2}}{2} + 1 + \frac{9}{11 2} arctan (\frac{x}{2}) - \frac{8}{11} ln ∣ x + 3 ∣ + C

Graph

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