integral from 0 to pi of xcos(x/2)

Lösung

\int_{0}^{π} x cos (\frac{x}{2}) d x

4 (\frac{π}{2} - 1)

Dezimale

2.28318 \dots

Schritte zur Lösung

\int_{0}^{π} x cos (\frac{x}{2}) d x

Wende U-Substitution an

= \int_{0}^{\frac{π}{2}} 4 u cos (u) d u

Entferne die Konstante:

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

= 4 \cdot \int_{0}^{\frac{π}{2}} u cos (u) d u

Wende die partielle Integration an

= 4 [u sin (u) - \int sin (u) d u]_{0}^{\frac{π}{2}}

\int sin (u) d u = - cos (u)

= 4 [u sin (u) - (- cos (u))]_{0}^{\frac{π}{2}}

Vereinfache

= 4 [u sin (u) + cos (u)]_{0}^{\frac{π}{2}}

Berechne die Grenzen:

\frac{π}{2} - 1

= 4 (\frac{π}{2} - 1)

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