integral from 0 to 1 of 3x(x^2-1)^3

Lösung

\int_{0}^{1} 3 x (x^{2} - 1)^{3} d x

- \frac{3}{8}

Dezimale

- 0.375

Schritte zur Lösung

\int_{0}^{1} 3 x (x^{2} - 1)^{3} d x

Entferne die Konstante:

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

= 3 \cdot \int_{0}^{1} x (x^{2} - 1)^{3} d x

Multipliziere aus

x (x^{2} - 1)^{3} : x^{7} - 3 x^{5} + 3 x^{3} - x

= 3 \cdot \int_{0}^{1} x^{7} - 3 x^{5} + 3 x^{3} - x 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

= 3 (\int_{0}^{1} x^{7} d x - \int_{0}^{1} 3 x^{5} d x + \int_{0}^{1} 3 x^{3} d x - \int_{0}^{1} x d x)

\int_{0}^{1} x^{7} d x = \frac{1}{8}

\int_{0}^{1} 3 x^{5} d x = \frac{1}{2}

\int_{0}^{1} 3 x^{3} d x = \frac{3}{4}

\int_{0}^{1} x d x = \frac{1}{2}

= 3 (\frac{1}{8} - \frac{1}{2} + \frac{3}{4} - \frac{1}{2})

Vereinfache

3 (\frac{1}{8} - \frac{1}{2} + \frac{3}{4} - \frac{1}{2}) : - \frac{3}{8}

= - \frac{3}{8}

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