integral from (7pi)/6 to (11pi)/6 of 1/2 ((-6sin(x))^2-(3)^2)

Lösung

\int_{\frac{7 π}{6}}^{\frac{11 π}{6}} \frac{1}{2} ((- 6 sin (x))^{2} - (3)^{2}) d x

\frac{6 π + 9 3}{2}

Dezimale

17.21900 \dots

Schritte zur Lösung

\int_{\frac{7 π}{6}}^{\frac{11 π}{6}} \frac{1}{2} ((- 6 sin (x))^{2} - 3^{2}) d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{\frac{7 π}{6}}^{\frac{11 π}{6}} (- 6 sin (x))^{2} - 3^{2} d x

Vereinfache

(- 6 sin (x))^{2} - 3^{2} : 36 sin^{2} (x) - 9

= \frac{1}{2} \cdot \int_{\frac{7 π}{6}}^{\frac{11 π}{6}} 36 sin^{2} (x) - 9 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

= \frac{1}{2} (\int_{\frac{7 π}{6}}^{\frac{11 π}{6}} 36 sin^{2} (x) d x - \int_{\frac{7 π}{6}}^{\frac{11 π}{6}} 9 d x)

\int_{\frac{7 π}{6}}^{\frac{11 π}{6}} 36 sin^{2} (x) d x = 18 (\frac{2 π}{3} + \frac{3}{2})

\int_{\frac{7 π}{6}}^{\frac{11 π}{6}} 9 d x = 6 π

= \frac{1}{2} (18 (\frac{2 π}{3} + \frac{3}{2}) - 6 π)

Vereinfache

\frac{1}{2} (18 (\frac{2 π}{3} + \frac{3}{2}) - 6 π) : \frac{6 π + 9 3}{2}

= \frac{6 π + 9 3}{2}

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