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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Wende die Potenzregel an

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

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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