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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vereinfache

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

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die Potenzregel an

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

Setze in

u = x^{\frac{2}{3}} + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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