integral from 0 to 3 of x^3sqrt(81-x^4)

Lösung

\int_{0}^{3} x^{3} 81 - x^{4} d x

\frac{243}{2}

Dezimale

121.5

Schritte zur Lösung

\int_{0}^{3} x^{3} 81 - x^{4} d x

Wende U-Substitution an

= \int_{81}^{0} - \frac{u}{4} d u

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

= - \int_{0}^{81} - \frac{u}{4} d u

Entferne die Konstante:

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

= - (- \frac{1}{4} \cdot \int_{0}^{81} u d u)

Wende die Potenzregel an

= - (- \frac{1}{4} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{81})

Vereinfache

= \frac{1}{4} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{81}

Berechne die Grenzen:

486

= \frac{1}{4} \cdot 486

Vereinfache

= \frac{243}{2}

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