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

Lösung

\int_{0}^{1} sin (\frac{x ^{2}}{2}) d x

2 (\frac{π}{2} § (\frac{2}{π} \frac{1}{2}) - \frac{π}{2} § (\frac{2}{π} \cdot 0))

Schritte zur Lösung

\int_{0}^{1} sin (\frac{x ^{2}}{2}) d x

Wende Exponentenregel an:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{x ^{2}}{2} = (\frac{x}{2})^{2}

= \int_{0}^{1} sin ((\frac{x}{2})^{2}) d x

Wende U-Substitution an

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

Entferne die Konstante:

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

= 2 \cdot \int_{0}^{\frac{1}{2}} sin (u^{2}) d u

Das ist ein nicht elementares Integral :

\int sin (x^{2}) d x = \frac{π}{2} § (\frac{2}{π} x)

= 2 [\frac{π}{2} § (\frac{2}{π} u)]_{0}^{\frac{1}{2}}

Berechne die Grenzen:

\frac{π}{2} § (\frac{2}{π} \frac{1}{2}) - \frac{π}{2} § (\frac{2}{π} \cdot 0)

= 2 (\frac{π}{2} § (\frac{2}{π} \frac{1}{2}) - \frac{π}{2} § (\frac{2}{π} \cdot 0))

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