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Beliebt Rechnen >

integral of 5/(x(x^5+4))

Lösung

\int \frac{5}{x ( x ^{5} + 4 )} d x

Lösung

\frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4 + C

Schritte zur Lösung

\int \frac{5}{x ( x ^{5} + 4 )} d x

Entferne die Konstante:

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

= 5 \cdot \int \frac{1}{x ( x ^{5} + 4 )} d x

Wende U-Substitution an

= 5 \cdot \int \frac{1}{5 u ( u + 4 )} d u

Entferne die Konstante:

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

= 5 \cdot \frac{1}{5} \cdot \int \frac{1}{u ( u + 4 )} d u

Ermittle den Partialbruch von

\frac{1}{u ( u + 4 )} : \frac{1}{4 u} - \frac{1}{4 ( u + 4 )}

= 5 \cdot \frac{1}{5} \cdot \int \frac{1}{4 u} - \frac{1}{4 ( u + 4 )} 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

= 5 \cdot \frac{1}{5} (\int \frac{1}{4 u} d u - \int \frac{1}{4 ( u + 4 )} d u)

\int \frac{1}{4 u} d u = \frac{1}{4} ln ∣ u ∣

\int \frac{1}{4 ( u + 4 )} d u = \frac{1}{4} ln ∣ u + 4 ∣

= 5 \cdot \frac{1}{5} (\frac{1}{4} ln ∣ u ∣ - \frac{1}{4} ln ∣ u + 4 ∣)

Setze in

u = x^{5}

ein

= 5 \cdot \frac{1}{5} (\frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4)

Vereinfache

5 \cdot \frac{1}{5} (\frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4) : \frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4

= \frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} ln x^{5} - \frac{1}{4} ln x^{5} + 4 + C

Graph

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