integral from 0 to 3 of 2xsqrt(9-x^2)

Lösung

\int_{0}^{3} 2 x 9 - x^{2} d x

18

Schritte zur Lösung

\int_{0}^{3} 2 x 9 - x^{2} d x

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{3} x 9 - x^{2} d x

Wende U-Substitution an

= 2 \cdot \int_{9}^{0} - \frac{u}{2} d u

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

= 2 (- \int_{0}^{9} - \frac{u}{2} d u)

Entferne die Konstante:

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

= 2 (- (- \frac{1}{2} \cdot \int_{0}^{9} u d u))

Wende die Potenzregel an

= 2 (- (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{9}))

Vereinfache

2 (- (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{9})) : [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{9}

= [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{9}

Berechne die Grenzen:

18

= 18

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