integral from 1 to 9 of x\sqrt[3]{1-x}

Lösung

\int_{1}^{9} x 3 1 - x d x

- \frac{468}{7}

Dezimale

- 66.85714 \dots

Schritte zur Lösung

\int_{1}^{9} x 3 1 - x d x

Wende U-Substitution an

= \int_{0}^{- 8} - (- u + 1) 3 u d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 8}^{0} - (- u + 1) 3 u d u

Entferne die Konstante:

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

= - (- \int_{- 8}^{0} (- u + 1) 3 u d u)

Multipliziere aus

(- u + 1) 3 u : - u^{\frac{4}{3}} + 3 u

= - (- \int_{- 8}^{0} - u^{\frac{4}{3}} + 3 u d u)

Wende die Summenregel an:

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

= - (- (- \int_{- 8}^{0} u^{\frac{4}{3}} d u + \int_{- 8}^{0} 3 u d u))

\int_{- 8}^{0} u^{\frac{4}{3}} d u = - \frac{384 3 - 1}{7}

\int_{- 8}^{0} 3 u d u = 12 3 - 1

= - (- (- (- \frac{384 3 - 1}{7}) + 12 3 - 1))

Vereinfache

- (- (- (- \frac{384 3 - 1}{7}) + 12 3 - 1)) : - \frac{468}{7}

= - \frac{468}{7}

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