integral from 1 to 3 of sqrt((x-1)(3-x))

Lösung

\int_{1}^{3} (x - 1) (3 - x) d x

\frac{π}{2}

Dezimale

1.57079 \dots

Schritte zur Lösung

\int_{1}^{3} (x - 1) (3 - x) d x

Wende U-Substitution an

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

Vervollständige das Quadrat

u (- u + 2) : - (u - 1)^{2} + 1

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

Wende U-Substitution an

= \int_{- 1}^{1} - v^{2} + 1 d v

Trigonometrische Substitution anwenden

= \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{2} (w) d w

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + cos ( 2 w )}{2} d w

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + cos (2 w) d w

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d w + \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 w) d w)

\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d w = π

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 w) d w = 0

= \frac{1}{2} (π + 0)

Vereinfache

\frac{1}{2} (π + 0) : \frac{π}{2}

= \frac{π}{2}

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