integral from pi/6 to pi/2 of 8cot^2(x)

Lösung

\int_{\frac{π}{6}}^{\frac{π}{2}} 8 cot^{2} (x) d x

8 (- \frac{π}{3} + 3)

Dezimale

5.47882 \dots

Schritte zur Lösung

\int_{\frac{π}{6}}^{\frac{π}{2}} 8 cot^{2} (x) d x

Entferne die Konstante:

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

= 8 \cdot \int_{\frac{π}{6}}^{\frac{π}{2}} cot^{2} (x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 8 \cdot \int_{\frac{π}{6}}^{\frac{π}{2}} - 1 + csc^{2} (x) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 8 (- \int_{\frac{π}{6}}^{\frac{π}{2}} 1 d x + \int_{\frac{π}{6}}^{\frac{π}{2}} csc^{2} (x) d x)

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

\int_{\frac{π}{6}}^{\frac{π}{2}} csc^{2} (x) d x = 3

= 8 (- \frac{π}{3} + 3)

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