integral of (x+3)/(2x^3-16)

Lösung

\int \frac{x + 3}{2 x ^{3} - 16} d x

\frac{1}{2} (\frac{5}{12} ln ∣ x - 2 ∣ + \frac{1}{24} (- 5 ln x^{2} + 2 x + 4 - 23 arctan (\frac{x + 1}{3}))) + C

Schritte zur Lösung

\int \frac{x + 3}{2 x ^{3} - 16} d x

\frac{x + 3}{2 x ^{3} - 16} = \frac{1}{2} \frac{x + 3}{x ^{3} - 8}

= \int \frac{1}{2} \cdot \frac{x + 3}{x ^{3} - 8} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{x + 3}{x ^{3} - 8} d x

Ermittle den Partialbruch von

\frac{x + 3}{x ^{3} - 8} : \frac{5}{12 ( x - 2 )} + \frac{- 5 x - 8}{12 ( x ^{2} + 2 x + 4 )}

= \frac{1}{2} \cdot \int \frac{5}{12 ( x - 2 )} + \frac{- 5 x - 8}{12 ( x ^{2} + 2 x + 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

= \frac{1}{2} (\int \frac{5}{12 ( x - 2 )} d x + \int \frac{- 5 x - 8}{12 ( x ^{2} + 2 x + 4 )} d x)

\int \frac{5}{12 ( x - 2 )} d x = \frac{5}{12} ln ∣ x - 2 ∣

\int \frac{- 5 x - 8}{12 ( x ^{2} + 2 x + 4 )} d x = \frac{1}{24} (- 5 ln x^{2} + 2 x + 4 - 23 arctan (\frac{x + 1}{3}))

= \frac{1}{2} (\frac{5}{12} ln ∣ x - 2 ∣ + \frac{1}{24} (- 5 ln x^{2} + 2 x + 4 - 23 arctan (\frac{x + 1}{3})))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (\frac{5}{12} ln ∣ x - 2 ∣ + \frac{1}{24} (- 5 ln x^{2} + 2 x + 4 - 23 arctan (\frac{x + 1}{3}))) + C

\int \frac{2 x ^{2}}{x ^{3} + 12} d x

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