integral of (4x)/(5(6x^2-7)^{2/3)}

Lösung

\int \frac{4 x}{5 ( 6 x ^{2} - 7 ) ^{\frac{2}{3}}} d x

\frac{1}{5} (6 x^{2} - 7)^{\frac{1}{3}} + C

Schritte zur Lösung

\int \frac{4 x}{5 ( 6 x ^{2} - 7 ) ^{\frac{2}{3}}} d x

Entferne die Konstante:

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

= \frac{4}{5} \cdot \int \frac{x}{( 6 x ^{2} - 7 ) ^{\frac{2}{3}}} d x

Wende U-Substitution an

= \frac{4}{5} \cdot \int \frac{1}{12 u ^{\frac{2}{3}}} d u

Entferne die Konstante:

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

= \frac{4}{5} \cdot \frac{1}{12} \cdot \int \frac{1}{u ^{\frac{2}{3}}} d u

Wende die Potenzregel an

= \frac{4}{5} \cdot \frac{1}{12} \cdot 3 u^{\frac{1}{3}}

Setze in

u = 6 x^{2} - 7

ein

= \frac{4}{5} \cdot \frac{1}{12} \cdot 3 (6 x^{2} - 7)^{\frac{1}{3}}

Vereinfache

\frac{4}{5} \cdot \frac{1}{12} \cdot 3 (6 x^{2} - 7)^{\frac{1}{3}} : \frac{1}{5} (6 x^{2} - 7)^{\frac{1}{3}}

= \frac{1}{5} (6 x^{2} - 7)^{\frac{1}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} (6 x^{2} - 7)^{\frac{1}{3}} + C

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