integral from 0 to 1/2 of (x^2)/(sqrt(1-x^2))

Lösung

\int_{0}^{\frac{1}{2}} \frac{x ^{2}}{1 - x ^{2}} d x

\frac{2 π - 3 3}{24}

Dezimale

0.04529 \dots

Schritte zur Lösung

\int_{0}^{\frac{1}{2}} \frac{x ^{2}}{1 - x ^{2}} d x

Trigonometrische Substitution anwenden

= \int_{0}^{\frac{π}{6}} sin^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{0}^{\frac{π}{6}} \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

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{6}} 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

= \frac{1}{2} (\int_{0}^{\frac{π}{6}} 1 d u - \int_{0}^{\frac{π}{6}} cos (2 u) d u)

\int_{0}^{\frac{π}{6}} 1 d u = \frac{π}{6}

\int_{0}^{\frac{π}{6}} cos (2 u) d u = \frac{3}{4}

= \frac{1}{2} (\frac{π}{6} - \frac{3}{4})

Vereinfache

\frac{1}{2} (\frac{π}{6} - \frac{3}{4}) : \frac{2 π - 3 3}{24}

= \frac{2 π - 3 3}{24}

\frac{d y}{d x} = 3 y - y^{2}

\int \frac{ln ( 7 x )}{x} d x

\frac{d}{d t} (- 3 t)

d er i v a t i v e f (x) = \frac{x}{5}

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