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Beliebt Rechnen >

integral of x^2(1+3x^3)^2

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

(1 + 3 u)^{2} : 1 + 6 u + 9 u^{2}

= \frac{1}{3} \cdot \int 1 + 6 u + 9 u^{2} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} (\int 1 d u + \int 6 u d u + \int 9 u^{2} d u)

\int 1 d u = u

\int 6 u d u = 3 u^{2}

\int 9 u^{2} d u = 3 u^{3}

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

Setze in

u = x^{3}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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