integral of (x^2+2)(x^3+6x+1)^{1/2}

Lösung

\int (x^{2} + 2) (x^{3} + 6 x + 1)^{\frac{1}{2}} d x

\frac{2}{9} (x^{3} + 6 x + 1)^{\frac{3}{2}} + C

Schritte zur Lösung

\int (x^{2} + 2) (x^{3} + 6 x + 1)^{\frac{1}{2}} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die Potenzregel an

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

Setze in

u = x^{3} + 6 x + 1

ein

= \frac{1}{3} \cdot \frac{2}{3} (x^{3} + 6 x + 1)^{\frac{3}{2}}

Vereinfache

\frac{1}{3} \cdot \frac{2}{3} (x^{3} + 6 x + 1)^{\frac{3}{2}} : \frac{2}{9} (x^{3} + 6 x + 1)^{\frac{3}{2}}

= \frac{2}{9} (x^{3} + 6 x + 1)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{9} (x^{3} + 6 x + 1)^{\frac{3}{2}} + C

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