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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x + 4}{x ^{3} + 4 x} : \frac{1}{x} + \frac{- x + 1}{x ^{2} + 4}

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Füge eine Konstante zur Lösung hinzu

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

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