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

Lösung

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

\frac{3 π + 2}{8}

Dezimale

1.42809 \dots

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

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

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

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

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

Vereinfache

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

= \frac{3 π + 2}{8}

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