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Beliebt Rechnen >

integral of (x^3+x)(x^4+2x^2+9)^{1/2}

Lösung

\int (x^{3} + x) (x^{4} + 2 x^{2} + 9)^{\frac{1}{2}} d x

Lösung

\frac{1}{6} (x^{4} + 2 x^{2} + 9)^{\frac{3}{2}} + C

Schritte zur Lösung

\int (x^{3} + x) (x^{4} + 2 x^{2} + 9)^{\frac{1}{2}} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die Potenzregel an

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

Setze in

u = x^{4} + 2 x^{2} + 9

ein

= \frac{1}{4} \cdot \frac{2}{3} (x^{4} + 2 x^{2} + 9)^{\frac{3}{2}}

Vereinfache

\frac{1}{4} \cdot \frac{2}{3} (x^{4} + 2 x^{2} + 9)^{\frac{3}{2}} : \frac{1}{6} (x^{4} + 2 x^{2} + 9)^{\frac{3}{2}}

= \frac{1}{6} (x^{4} + 2 x^{2} + 9)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} (x^{4} + 2 x^{2} + 9)^{\frac{3}{2}} + C

Graph

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