integral from 0 to 2pi of (1-|cos(x)|)

Lösung

\int_{0}^{2 π} (1 - ∣ cos (x) ∣) d x

2 π - 4

Dezimale

2.28318 \dots

Schritte zur Lösung

\int_{0}^{2 π} 1 - ∣ cos (x) ∣ d x

Entferne die Extremwerte

= \int_{0}^{\frac{π}{2}} 1 - cos (x) d x + \int_{\frac{π}{2}}^{\frac{3 π}{2}} 1 - (- cos (x)) d x + \int_{\frac{3 π}{2}}^{2 π} 1 - cos (x) d x

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

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

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

= \frac{π}{2} - 1 + π - 2 + 2 π - \frac{3 π}{2} - 1

Vereinfache

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

= 2 π - 4