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Beliebt Rechnen >

integral of x/(\sqrt[3]{1+x^2)}

Lösung

\int \frac{x}{3 1 + x ^{2}} d x

Lösung

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

Schritte zur Lösung

\int \frac{x}{3 1 + x ^{2}} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die Potenzregel an

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

Setze in

u = 1 + x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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