integral from 0 to 3 of x^2sqrt(9-x^2)

Lösung

\int_{0}^{3} x^{2} 9 - x^{2} d x

\frac{81 π}{16}

Dezimale

15.90431 \dots

Schritte zur Lösung

\int_{0}^{3} x^{2} 9 - x^{2} d x

Trigonometrische Substitution anwenden

= \int_{0}^{\frac{π}{2}} 81 sin^{2} (u) cos^{2} (u) d u

Entferne die Konstante:

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

= 81 \cdot \int_{0}^{\frac{π}{2}} sin^{2} (u) cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 81 \cdot \int_{0}^{\frac{π}{2}} \frac{1 - cos ( 4 u )}{8} d u

Entferne die Konstante:

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

= 81 \cdot \frac{1}{8} \cdot \int_{0}^{\frac{π}{2}} 1 - cos (4 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

= 81 \cdot \frac{1}{8} (\int_{0}^{\frac{π}{2}} 1 d u - \int_{0}^{\frac{π}{2}} cos (4 u) d u)

\int_{0}^{\frac{π}{2}} 1 d u = \frac{π}{2}

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

= 81 \cdot \frac{1}{8} (\frac{π}{2} - 0)

Vereinfache

81 \cdot \frac{1}{8} (\frac{π}{2} - 0) : \frac{81 π}{16}

= \frac{81 π}{16}

n = 0 \sum \infty (\frac{n x}{6})^{n}

x \to + 2 lim (2 - x)

d er i v a t i v e f (x) = (2 t - 1) (3 t - 4)^{- 1}

\frac{\partial}{\partial x} (x^{3} + 2 x y + y^{2})

\frac{d}{d x} (0.015 + x)