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

Lösung

\int_{0}^{a} a^{2} - x^{2} d x

\frac{π}{4} a^{2}

Schritte zur Lösung

\int_{0}^{a} a^{2} - x^{2} d x

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= a^{2} \cdot \int_{0}^{\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

= a^{2} \frac{1}{2} \cdot \int_{0}^{\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

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

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

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

= a^{2} \frac{1}{2} (\frac{π}{2} + 0)

Vereinfache

a^{2} \frac{1}{2} (\frac{π}{2} + 0) : \frac{π}{4} a^{2}

= \frac{π}{4} a^{2}

\frac{\partial}{\partial x} (e^{2 x^{2}})

d er i v a t i v e f (x) = 3 x

d er i v a t i v e x^{2} - 5 x - 1

\frac{d}{d x} (3 x cos (3 x))

d er i v a t i v e f (x) = - 5 cot (x) + 3 sec (x)