integral of 3/(x(x^4+8))

Lösung

\int \frac{3}{x ( x ^{4} + 8 )} d x

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

Schritte zur Lösung

\int \frac{3}{x ( x ^{4} + 8 )} d x

Entferne die Konstante:

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

= 3 \cdot \int \frac{1}{x ( x ^{4} + 8 )} d x

Wende U-Substitution an

= 3 \cdot \int \frac{1}{4 u ( u + 8 )} d u

Entferne die Konstante:

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

= 3 \cdot \frac{1}{4} \cdot \int \frac{1}{u ( u + 8 )} d u

Ermittle den Partialbruch von

\frac{1}{u ( u + 8 )} : \frac{1}{8 u} - \frac{1}{8 ( u + 8 )}

= 3 \cdot \frac{1}{4} \cdot \int \frac{1}{8 u} - \frac{1}{8 ( u + 8 )} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 \cdot \frac{1}{4} (\int \frac{1}{8 u} d u - \int \frac{1}{8 ( u + 8 )} d u)

\int \frac{1}{8 u} d u = \frac{1}{8} ln ∣ u ∣

\int \frac{1}{8 ( u + 8 )} d u = \frac{1}{8} ln ∣ u + 8 ∣

= 3 \cdot \frac{1}{4} (\frac{1}{8} ln ∣ u ∣ - \frac{1}{8} ln ∣ u + 8 ∣)

Setze in

u = x^{4}

ein

= 3 \cdot \frac{1}{4} (\frac{1}{8} ln x^{4} - \frac{1}{8} ln x^{4} + 8)

Vereinfache

3 \cdot \frac{1}{4} (\frac{1}{8} ln x^{4} - \frac{1}{8} ln x^{4} + 8) : \frac{3}{4} (\frac{1}{2} ln (x) - \frac{1}{8} ln x^{4} + 8)

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

Füge eine Konstante zur Lösung hinzu

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

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