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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{1}{4 u ^{\frac{1}{3}}} d u

Entferne die Konstante:

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

= \frac{1}{4} \cdot \int \frac{1}{u ^{\frac{1}{3}}} d u

Wende die Potenzregel an

= \frac{1}{4} \cdot \frac{3}{2} u^{\frac{2}{3}}

Setze in

u = 1 + x^{4}

ein

= \frac{1}{4} \cdot \frac{3}{2} (1 + x^{4})^{\frac{2}{3}}

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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