integral of (x^4+7)/(x^3+9x)

Lösung

\int \frac{x ^{4} + 7}{x ^{3} + 9 x} d x

\frac{x ^{2}}{2} - \frac{44}{9} ln x^{2} + 9 + \frac{7}{9} ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{x ^{4} + 7}{x ^{3} + 9 x} d x

Lange Division

\frac{x ^{4} + 7}{x ^{3} + 9 x} : x + \frac{- 9 x ^{2} + 7}{x ^{3} + 9 x}

= \int x + \frac{- 9 x ^{2} + 7}{x ^{3} + 9 x} 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 x d x + \int \frac{- 9 x ^{2} + 7}{x ^{3} + 9 x} d x

\int x d x = \frac{x ^{2}}{2}

\int \frac{- 9 x ^{2} + 7}{x ^{3} + 9 x} d x = - \frac{44}{9} ln x^{2} + 9 + \frac{7}{9} ln ∣ x ∣

= \frac{x ^{2}}{2} - \frac{44}{9} ln x^{2} + 9 + \frac{7}{9} ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - \frac{44}{9} ln x^{2} + 9 + \frac{7}{9} ln ∣ x ∣ + C

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