integral from pi/3 to pi of xcos(2x)

Lösung

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

\frac{3 3 - 2 π}{8 3}

Dezimale

- 0.07844 \dots

Schritte zur Lösung

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

Wende U-Substitution an

= \int_{\frac{2 π}{3}}^{2 π} \frac{u cos ( u )}{4} d u

Entferne die Konstante:

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

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

Wende die partielle Integration an

= \frac{1}{4} [u sin (u) - \int sin (u) d u]_{\frac{2 π}{3}}^{2 π}

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

= \frac{1}{4} [u sin (u) - (- cos (u))]_{\frac{2 π}{3}}^{2 π}

Vereinfache

= \frac{1}{4} [u sin (u) + cos (u)]_{\frac{2 π}{3}}^{2 π}

Berechne die Grenzen:

\frac{3}{2} - \frac{π}{3}

= \frac{1}{4} (\frac{3}{2} - \frac{π}{3})

Vereinfache

= \frac{3 3 - 2 π}{8 3}

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