integral from pi/2 to pi of (1+cos(x))/x

Lösung

\int_{\frac{π}{2}}^{π} \frac{1 + cos ( x )}{x} d x

ln (2) + Ci (π) - Ci (\frac{π}{2})

Schritte zur Lösung

\int_{\frac{π}{2}}^{π} \frac{1 + cos ( x )}{x} d x

Multipliziere aus

\frac{1 + cos ( x )}{x} : \frac{1}{x} + \frac{cos ( x )}{x}

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

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

\int_{\frac{π}{2}}^{π} \frac{1}{x} d x = ln (2)

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

= ln (2) + Ci (π) - Ci (\frac{π}{2})

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