integral from 0.1 to 3 of xcos(1/x)

Lösung

\int_{0.1}^{3} x cos (\frac{1}{x}) d x

0.5 Ci (\frac{1}{3}) + 3.73850 \dots - 0.5 Ci (\frac{1}{0.1})

Schritte zur Lösung

\int_{0.1}^{3} x cos (\frac{1}{x}) d x

Wende die partielle Integration an

= [0.5 x^{2} cos (\frac{1}{x}) - \int \frac{1}{2} sin (\frac{1}{x}) d x]_{0.1}^{3}

\int \frac{1}{2} sin (\frac{1}{x}) d x = \frac{1}{2} (x sin (\frac{1}{x}) - Ci (\frac{1}{x}))

= [0.5 x^{2} cos (\frac{1}{x}) - \frac{1}{2} (x sin (\frac{1}{x}) - Ci (\frac{1}{x}))]_{0.1}^{3}

Vereinfache

= [0.5 x^{2} cos (\frac{1}{x}) - 0.5 (x sin (\frac{1}{x}) - Ci (\frac{1}{x}))]_{0.1}^{3}

Berechne die Grenzen:

0.5 Ci (\frac{1}{3}) + 3.73850 \dots - 0.5 Ci (\frac{1}{0.1})

= 0.5 Ci (\frac{1}{3}) + 3.73850 \dots - 0.5 Ci (\frac{1}{0.1})

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