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

Lösung

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

8

Schritte zur Lösung

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

Wende U-Substitution an

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

\int_{1}^{2} u^{2} d u = \frac{7}{3}

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

= 3 \cdot 2 (\frac{7}{3} - 1)

Vereinfache

3 \cdot 2 (\frac{7}{3} - 1) : 8

= 8

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