integral from-1 to 1 of 4x^2(-x^3-1)^3

Lösung

\int_{- 1}^{1} 4 x^{2} (- x^{3} - 1)^{3} d x

- \frac{16}{3}

Dezimale

- 5.33333 \dots

Schritte zur Lösung

\int_{- 1}^{1} 4 x^{2} (- x^{3} - 1)^{3} d x

Entferne die Konstante:

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

= 4 \cdot \int_{- 1}^{1} x^{2} (- x^{3} - 1)^{3} d x

Multipliziere aus

x^{2} (- x^{3} - 1)^{3} : - x^{11} - 3 x^{8} - 3 x^{5} - x^{2}

= 4 \cdot \int_{- 1}^{1} - x^{11} - 3 x^{8} - 3 x^{5} - x^{2} d x

Wende die Summenregel an:

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

= 4 (- \int_{- 1}^{1} x^{11} d x - \int_{- 1}^{1} 3 x^{8} d x - \int_{- 1}^{1} 3 x^{5} d x - \int_{- 1}^{1} x^{2} d x)

\int_{- 1}^{1} x^{11} d x = 0

\int_{- 1}^{1} 3 x^{8} d x = \frac{2}{3}

\int_{- 1}^{1} 3 x^{5} d x = 0

\int_{- 1}^{1} x^{2} d x = \frac{2}{3}

= 4 (- 0 - \frac{2}{3} - 0 - \frac{2}{3})

Vereinfache

4 (- 0 - \frac{2}{3} - 0 - \frac{2}{3}) : - \frac{16}{3}

= - \frac{16}{3}

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