integral from 0 to 1 of x(2(x+1))/3

Lösung

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

\frac{5}{9}

Dezimale

0.55555 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Multipliziere aus

x (x + 1) : x^{2} + x

= \frac{2}{3} \cdot \int_{0}^{1} x^{2} + 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

= \frac{2}{3} (\int_{0}^{1} x^{2} d x + \int_{0}^{1} x d x)

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

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

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

Vereinfache

\frac{2}{3} (\frac{1}{3} + \frac{1}{2}) : \frac{5}{9}

= \frac{5}{9}

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