What is the integral from 0.1 to 3 of xcos(1/x) ?

The integral from 0.1 to 3 of xcos(1/x) is 0.5\Ci(1/3)+3.73850…-0.5\Ci(1/(0.1))

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

\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})

Solution steps

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

Apply Integration By Parts

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

= [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}

Simplify

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

Compute the boundaries:

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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What is the integral from 0.1 to 3 of xcos(1/x) ?
The integral from 0.1 to 3 of xcos(1/x) is 0.5\Ci(1/3)+3.73850…-0.5\Ci(1/(0.1))