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integral from 0 to x of sin(8t^2)

Lösung

\int_{0}^{x} sin (8 t^{2}) d t

\frac{1}{2 2} (\frac{π}{2} § (\frac{2}{π} \cdot 22 x) - \frac{π}{2} § (\frac{2}{π} \cdot 0))

Schritte zur Lösung

\int_{0}^{x} sin (8 t^{2}) d t

Wende integrale Substitution an

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

Entferne die Konstante:

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

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

Wende integrale Substitution an

= \frac{1}{2} \cdot \int_{0}^{4 x} \frac{1}{2} sin (\frac{v ^{2}}{2}) d v

Entferne die Konstante:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int_{0}^{4 x} sin (\frac{v ^{2}}{2}) d v

Wende Exponentenregel an:

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

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int_{0}^{4 x} sin ((\frac{v}{2})^{2}) d v

Wende U-Substitution an

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int_{0}^{22 x} sin (w^{2}) 2 d w

Entferne die Konstante:

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

= \frac{1}{2} \cdot \frac{1}{2} 2 \cdot \int_{0}^{22 x} sin (w^{2}) d w

Das ist ein nicht elementares Integral :

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

= \frac{1}{2} \cdot \frac{1}{2} 2 [\frac{π}{2} § (\frac{2}{π} w)]_{0}^{22 x}

Vereinfache

\frac{1}{2} \cdot \frac{1}{2} 2 [\frac{π}{2} § (\frac{2}{π} w)]_{0}^{22 x} : \frac{1}{2 2} [\frac{π}{2} § (\frac{2}{π} w)]_{0}^{22 x}

= \frac{1}{2 2} [\frac{π}{2} § (\frac{2}{π} w)]_{0}^{22 x}

Berechne die Grenzen:

\frac{π}{2} § (\frac{2}{π} \cdot 22 x) - \frac{π}{2} § (\frac{2}{π} \cdot 0)

= \frac{1}{2 2} (\frac{π}{2} § (\frac{2}{π} \cdot 22 x) - \frac{π}{2} § (\frac{2}{π} \cdot 0))

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