integral from pi to 0 of-sin^2(t)

Lösung

\int_{π}^{0} - sin^{2} (t) d t

\frac{π}{2}

Dezimale

1.57079 \dots

Schritte zur Lösung

\int_{π}^{0} - sin^{2} (t) d t

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{0}^{π} - sin^{2} (t) d t

Entferne die Konstante:

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

= - (- \int_{0}^{π} sin^{2} (t) d t)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= - (- \int_{0}^{π} \frac{1 - cos ( 2 t )}{2} d t)

Entferne die Konstante:

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

= - (- \frac{1}{2} \cdot \int_{0}^{π} 1 - cos (2 t) d t)

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_{0}^{π} 1 d t - \int_{0}^{π} cos (2 t) d t))

\int_{0}^{π} 1 d t = π

\int_{0}^{π} cos (2 t) d t = 0

= - (- \frac{1}{2} (π - 0))

Vereinfache

- (- \frac{1}{2} (π - 0)) : \frac{π}{2}

= \frac{π}{2}

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