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

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{3 ( 4 u + 1 ) ^{\frac{1}{2}}}{4} d u

Entferne die Konstante:

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

= \frac{3}{4} \cdot \int (4 u + 1)^{\frac{1}{2}} d u

Wende U-Substitution an

= \frac{3}{4} \cdot \int \frac{v ^{\frac{1}{2}}}{4} d v

Entferne die Konstante:

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

= \frac{3}{4} \cdot \frac{1}{4} \cdot \int v^{\frac{1}{2}} d v

Wende die Potenzregel an

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

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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