integral from 0 to 1 of (x-3/4)^23x^2

Lösung

\int_{0}^{1} (x - \frac{3}{4})^{2} 3 x^{2} d x

\frac{3}{80}

Dezimale

0.0375

Schritte zur Lösung

\int_{0}^{1} (x - \frac{3}{4})^{2} \cdot 3 x^{2} 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 - \frac{3}{4})^{2} x^{2} d x

Multipliziere aus

(x - \frac{3}{4})^{2} x^{2} : x^{4} - \frac{3 x ^{3}}{2} + \frac{9 x ^{2}}{16}

= 3 \cdot \int_{0}^{1} x^{4} - \frac{3 x ^{3}}{2} + \frac{9 x ^{2}}{16} 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^{4} d x - \int_{0}^{1} \frac{3 x ^{3}}{2} d x + \int_{0}^{1} \frac{9 x ^{2}}{16} d x)

\int_{0}^{1} x^{4} d x = \frac{1}{5}

\int_{0}^{1} \frac{3 x ^{3}}{2} d x = \frac{3}{8}

\int_{0}^{1} \frac{9 x ^{2}}{16} d x = \frac{3}{16}

= 3 (\frac{1}{5} - \frac{3}{8} + \frac{3}{16})

Vereinfache

3 (\frac{1}{5} - \frac{3}{8} + \frac{3}{16}) : \frac{3}{80}

= \frac{3}{80}

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

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