integral from 0 to pi/2 of x/(1+cos(x))

Lösung

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

\frac{π}{2} - ln (2)

Dezimale

0.87764 \dots

Schritte zur Lösung

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

Wende die partielle Integration an

= [x tan (\frac{x}{2}) - \int tan (\frac{x}{2}) d x]_{0}^{\frac{π}{2}}

\int tan (\frac{x}{2}) d x = - 2 ln cos (\frac{x}{2})

= [x tan (\frac{x}{2}) - (- 2 ln cos (\frac{x}{2}))]_{0}^{\frac{π}{2}}

Vereinfache

= [x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2})]_{0}^{\frac{π}{2}}

Berechne die Grenzen:

\frac{π}{2} - ln (2)

= \frac{π}{2} - ln (2)

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