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

Lösung

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

\frac{3 ( 4 π - 3 3 )}{8}

Dezimale

2.76383 \dots

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

= \int_{\frac{π}{6}}^{\frac{π}{2}} 9 cos^{2} (u) d u

Entferne die Konstante:

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

= 9 \cdot \int_{\frac{π}{6}}^{\frac{π}{2}} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 9 \cdot \int_{\frac{π}{6}}^{\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

= 9 \cdot \frac{1}{2} \cdot \int_{\frac{π}{6}}^{\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

= 9 \cdot \frac{1}{2} (\int_{\frac{π}{6}}^{\frac{π}{2}} 1 d u + \int_{\frac{π}{6}}^{\frac{π}{2}} cos (2 u) d u)

\int_{\frac{π}{6}}^{\frac{π}{2}} 1 d u = \frac{π}{3}

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

= 9 \cdot \frac{1}{2} (\frac{π}{3} - \frac{3}{4})

Vereinfache

9 \cdot \frac{1}{2} (\frac{π}{3} - \frac{3}{4}) : \frac{3 ( 4 π - 3 3 )}{8}

= \frac{3 ( 4 π - 3 3 )}{8}

\int_{0}^{3} \frac{1}{x ^{2} - x - 2} d x

x \to 3 lim (3 x^{2} - 2 x + 5)

\int \frac{sin ( x )}{cos ( x )} d x

\int 2 (3 x + 4)^{- 4} d x

t an g e n t f (x) = \frac{x}{x + 1}, (1, 2)