integral of (x^4)/((1+x^5)^{2/3)}

Lösung

\int \frac{x ^{4}}{( 1 + x ^{5} ) ^{\frac{2}{3}}} d x

\frac{3}{5} (1 + x^{5})^{\frac{1}{3}} + C

Schritte zur Lösung

\int \frac{x ^{4}}{( 1 + x ^{5} ) ^{\frac{2}{3}}} d x

Wende U-Substitution an

= \int \frac{1}{5 u ^{\frac{2}{3}}} d u

Entferne die Konstante:

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

= \frac{1}{5} \cdot \int \frac{1}{u ^{\frac{2}{3}}} d u

Wende die Potenzregel an

= \frac{1}{5} \cdot 3 u^{\frac{1}{3}}

Setze in

u = 1 + x^{5}

ein

= \frac{1}{5} \cdot 3 (1 + x^{5})^{\frac{1}{3}}

Vereinfache

\frac{1}{5} \cdot 3 (1 + x^{5})^{\frac{1}{3}} : \frac{3}{5} (1 + x^{5})^{\frac{1}{3}}

= \frac{3}{5} (1 + x^{5})^{\frac{1}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{5} (1 + x^{5})^{\frac{1}{3}} + C

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