integral from 0 to 5/2 of sqrt(25-x^2)

Lösung

\int_{0}^{\frac{5}{2}} 25 - x^{2} d x

\frac{25 ( 2 π + 3 3 )}{24}

Dezimale

11.95764 \dots

Schritte zur Lösung

\int_{0}^{\frac{5}{2}} 25 - x^{2} d x

Trigonometrische Substitution anwenden

= \int_{0}^{\frac{π}{6}} 25 cos^{2} (u) d u

Entferne die Konstante:

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

= 25 \cdot \int_{0}^{\frac{π}{6}} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 25 \cdot \int_{0}^{\frac{π}{6}} \frac{1 + cos ( 2 u )}{2} d u

Entferne die Konstante:

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

= 25 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{6}} 1 + cos (2 u) 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

= 25 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{6}} 1 d u + \int_{0}^{\frac{π}{6}} cos (2 u) d u)

\int_{0}^{\frac{π}{6}} 1 d u = \frac{π}{6}

\int_{0}^{\frac{π}{6}} cos (2 u) d u = \frac{3}{4}

= 25 \cdot \frac{1}{2} (\frac{π}{6} + \frac{3}{4})

Vereinfache

25 \cdot \frac{1}{2} (\frac{π}{6} + \frac{3}{4}) : \frac{25 ( 2 π + 3 3 )}{24}

= \frac{25 ( 2 π + 3 3 )}{24}

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