integral from pi/4 to pi/3 of sin^2(2x)

Lösung

\int_{\frac{π}{4}}^{\frac{π}{3}} sin^{2} (2 x) d x

\frac{2 π + 3 3}{48}

Dezimale

0.23915 \dots

Schritte zur Lösung

\int_{\frac{π}{4}}^{\frac{π}{3}} sin^{2} (2 x) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{1}{2} (1 - cos (4 x)) 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{π}{4}}^{\frac{π}{3}} 1 - cos (4 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

= \frac{1}{2} (\int_{\frac{π}{4}}^{\frac{π}{3}} 1 d x - \int_{\frac{π}{4}}^{\frac{π}{3}} cos (4 x) d x)

\int_{\frac{π}{4}}^{\frac{π}{3}} 1 d x = \frac{π}{12}

\int_{\frac{π}{4}}^{\frac{π}{3}} cos (4 x) d x = - \frac{3}{8}

= \frac{1}{2} (\frac{π}{12} - (- \frac{3}{8}))

Vereinfache

\frac{1}{2} (\frac{π}{12} - (- \frac{3}{8})) : \frac{2 π + 3 3}{48}

= \frac{2 π + 3 3}{48}

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