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

Lösung

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

2 (- \frac{π}{3} + 3)

Dezimale

1.36970 \dots

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

\frac{u ^{2}}{1 + u ^{2}} = - \frac{1}{1 + u ^{2}} + 1

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

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

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

\int_{0}^{3} 1 d u = 3

= 2 (- \frac{π}{3} + 3)

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