integral from 0 to 8 of 1/(v^{1/3)+1}

Lösung

\int_{0}^{8} \frac{1}{v ^{\frac{1}{3}} + 1} d v

3 ln (3)

Dezimale

3.29583 \dots

Schritte zur Lösung

\int_{0}^{8} \frac{1}{v ^{\frac{1}{3}} + 1} d v

Wende U-Substitution an

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

Entferne die Konstante:

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

= 3 \cdot \int_{0}^{2} \frac{u ^{2}}{u + 1} d u

Wende U-Substitution an

= 3 \cdot \int_{1}^{3} \frac{( w - 1 ) ^{2}}{w} d w

Multipliziere aus

\frac{( w - 1 ) ^{2}}{w} : w - 2 + \frac{1}{w}

= 3 \cdot \int_{1}^{3} w - 2 + \frac{1}{w} d w

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (\int_{1}^{3} w d w - \int_{1}^{3} 2 d w + \int_{1}^{3} \frac{1}{w} d w)

\int_{1}^{3} w d w = 4

\int_{1}^{3} 2 d w = 4

\int_{1}^{3} \frac{1}{w} d w = ln (3)

= 3 (4 - 4 + ln (3))

Vereinfache

= 3 ln (3)

\int_{0}^{2} π (x^{2} - 4)^{2} d x

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