integral from 0 to 8 of sqrt(1+x^{-2/3)}

Lösung

\int_{0}^{8} 1 + x^{- \frac{2}{3}} d x

55 - 1

Dezimale

10.18033 \dots

Schritte zur Lösung

\int_{0}^{8} 1 + x^{- \frac{2}{3}} d x

Vereinfache

1 + x^{- \frac{2}{3}} : \frac{x ^{\frac{2}{3}} + 1}{x ^{\frac{2}{3}}}

= \int_{0}^{8} \frac{x ^{\frac{2}{3}} + 1}{x ^{\frac{2}{3}}} d x

Wende U-Substitution an

= \int_{1}^{5} \frac{3 u ^{\frac{1}{2}}}{2} d u

Entferne die Konstante:

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

= \frac{3}{2} \cdot \int_{1}^{5} u^{\frac{1}{2}} d u

Wende die Potenzregel an

= \frac{3}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{5}

Berechne die Grenzen:

\frac{10 5}{3} - \frac{2}{3}

= \frac{3}{2} (\frac{10 5}{3} - \frac{2}{3})

Vereinfache

= 55 - 1

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