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

Lösung

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

\frac{1}{3} - \frac{3}{8}

Dezimale

0.11682 \dots

Schritte zur Lösung

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

Wende U-Substitution an

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

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

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

Entferne die Konstante:

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

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

Wende die Potenzregel an

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

Vereinfache

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

Berechne die Grenzen:

\frac{1}{3} - \frac{3}{8}

= \frac{1}{3} - \frac{3}{8}

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