integral from 0 to 5 of 15xsqrt(x+4)

Lösung

\int_{0}^{5} 15 x x + 4 d x

506

Schritte zur Lösung

\int_{0}^{5} 15 x x + 4 d x

Entferne die Konstante:

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

= 15 \cdot \int_{0}^{5} x x + 4 d x

Wende U-Substitution an

= 15 \cdot \int_{4}^{9} (u - 4) u d u

Multipliziere aus

(u - 4) u : u^{\frac{3}{2}} - 4 u

= 15 \cdot \int_{4}^{9} u^{\frac{3}{2}} - 4 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

= 15 (\int_{4}^{9} u^{\frac{3}{2}} d u - \int_{4}^{9} 4 u d u)

\int_{4}^{9} u^{\frac{3}{2}} d u = \frac{422}{5}

\int_{4}^{9} 4 u d u = \frac{152}{3}

= 15 (\frac{422}{5} - \frac{152}{3})

Vereinfache

15 (\frac{422}{5} - \frac{152}{3}) : 506

= 506

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