integral from 0 to 8 of (3x)/(sqrt(1+x))

Lösung

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

40

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Entferne die Konstante:

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

= 3 \cdot 2 \cdot \int_{1}^{3} u^{2} - 1 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

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

\int_{1}^{3} u^{2} d u = \frac{26}{3}

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

= 3 \cdot 2 (\frac{26}{3} - 2)

Vereinfache

3 \cdot 2 (\frac{26}{3} - 2) : 40

= 40

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