integral from 0 to 1 of 2xsqrt(1-x^2)

Lösung

\int_{0}^{1} 2 x 1 - x^{2} d x

\frac{2}{3}

Dezimale

0.66666 \dots

Schritte zur Lösung

\int_{0}^{1} 2 x 1 - x^{2} d x

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{1} x 1 - x^{2} d x

Wende U-Substitution an

= 2 \cdot \int_{1}^{0} - u^{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

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

Entferne die Konstante:

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

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

Wende die Potenzregel an

= 2 (- (- [\frac{u ^{3}}{3}]_{0}^{1}))

Vereinfache

= 2 [\frac{u ^{3}}{3}]_{0}^{1}

Berechne die Grenzen:

\frac{1}{3}

= 2 \cdot \frac{1}{3}

Vereinfache

= \frac{2}{3}

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