integral from 0 to 8 of xsqrt(8-x)

Lösung

\int_{0}^{8} x 8 - x d x

\frac{512 2}{15}

Dezimale

48.27182 \dots

Schritte zur Lösung

\int_{0}^{8} x 8 - x d x

Wende U-Substitution an

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

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

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

Entferne die Konstante:

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

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

Multipliziere aus

(- u + 8) u : - u^{\frac{3}{2}} + 8 u

= - (- \int_{0}^{8} - u^{\frac{3}{2}} + 8 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_{0}^{8} u^{\frac{3}{2}} d u + \int_{0}^{8} 8 u d u))

\int_{0}^{8} u^{\frac{3}{2}} d u = \frac{256 2}{5}

\int_{0}^{8} 8 u d u = \frac{256 2}{3}

= - (- (- \frac{256 2}{5} + \frac{256 2}{3}))

Vereinfache

- (- (- \frac{256 2}{5} + \frac{256 2}{3})) : \frac{512 2}{15}

= \frac{512 2}{15}

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