integral of (8x^2+5x+77)/((x+1)(x^2+9))

Lösung

\int \frac{8 x ^{2} + 5 x + 77}{( x + 1 ) ( x ^{2} + 9 )} d x

8 ln ∣ x + 1 ∣ + \frac{5}{3} arctan (\frac{x}{3}) + C

Schritte zur Lösung

\int \frac{8 x ^{2} + 5 x + 77}{( x + 1 ) ( x ^{2} + 9 )} d x

Ermittle den Partialbruch von

\frac{8 x ^{2} + 5 x + 77}{( x + 1 ) ( x ^{2} + 9 )} : \frac{8}{x + 1} + \frac{5}{x ^{2} + 9}

= \int \frac{8}{x + 1} + \frac{5}{x ^{2} + 9} 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 + 1} d x + \int \frac{5}{x ^{2} + 9} d x

\int \frac{8}{x + 1} d x = 8 ln ∣ x + 1 ∣

\int \frac{5}{x ^{2} + 9} d x = \frac{5}{3} arctan (\frac{x}{3})

= 8 ln ∣ x + 1 ∣ + \frac{5}{3} arctan (\frac{x}{3})

Füge eine Konstante zur Lösung hinzu

= 8 ln ∣ x + 1 ∣ + \frac{5}{3} arctan (\frac{x}{3}) + C

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