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

Lösung

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

\frac{π}{8} a^{4}

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

= \int_{- \frac{π}{2}}^{\frac{π}{2}} a^{4} 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

= a^{4} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{2} (u) cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= a^{4} \cdot \int_{- \frac{π}{2}}^{\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

= a^{4} \frac{1}{8} \cdot \int_{- \frac{π}{2}}^{\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

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

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

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

= a^{4} \frac{1}{8} (π - 0)

Vereinfache

a^{4} \frac{1}{8} (π - 0) : \frac{π}{8} a^{4}

= \frac{π}{8} a^{4}

\int_{0}^{1} (x - x^{2}) x d x

\int_{0}^{3} 4 x e^{- x} d x

\int_{0}^{\frac{π}{2}} x^{2} y cos (x y^{2}) d x

\int_{\frac{1}{2}}^{\frac{e}{2}} x ln (2 x) d x

\int_{0}^{π} cos^{3} (5 x) d x