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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

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

= 3 \cdot \frac{1}{3} (u (u + 1)^{\frac{1}{2}} - \frac{1}{3} (u - 2) (u + 1)^{\frac{1}{2}})

Setze in

u = x^{3}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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