integral of 1/(x^{4/5)(1+x^{1/5})}

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{5}{4 u ^{\frac{3}{4}} ( u ^{\frac{1}{4}} + 1 )} d u

Entferne die Konstante:

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

= \frac{5}{4} \cdot \int \frac{1}{u ^{\frac{3}{4}} ( u ^{\frac{1}{4}} + 1 )} d u

Wende U-Substitution an

= \frac{5}{4} \cdot \int \frac{4}{v} d v

Entferne die Konstante:

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

= \frac{5}{4} \cdot 4 \cdot \int \frac{1}{v} d v

Nutze das gemeinsame Integral :

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{5}{4} \cdot 4 ln ∣ v ∣

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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