integral from-R to R of sqrt(R^2-x^2)

Lösung

\int_{- R}^{R} R^{2} - x^{2} d x

\frac{π}{2} R^{2}

Schritte zur Lösung

\int_{- R}^{R} R^{2} - x^{2} d x

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

= R^{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= R^{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} \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

= R^{2} \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} 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

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

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

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

= R^{2} \frac{1}{2} (π + 0)

Vereinfache

R^{2} \frac{1}{2} (π + 0) : \frac{π}{2} R^{2}

= \frac{π}{2} R^{2}

\frac{\partial}{\partial y} (x e^{- \frac{y}{x}})

\int \frac{sec ( x ) tan ( x )}{1 + sec ( x )} d x

\frac{d}{d x} (\frac{1}{2 ^{x}})

x \to 0 lim (\frac{tan ( 10 x )}{x})

d er i v a t i v e 5^{x^{5}}