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

Lösung

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

\frac{1}{6}

Dezimale

0.16666 \dots

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Multipliziere aus

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

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

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

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

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

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

Vereinfache

\frac{1}{2} (\frac{1}{3} - 1 + 1) : \frac{1}{6}

= \frac{1}{6}

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