What is the integral from 0 to 1 of (cos(1/x))/(x^2) ?

The integral from 0 to 1 of (cos(1/x))/(x^2) is diverges

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

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

diverges

Solution steps

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

Apply Integration By Parts

= [\frac{cos ( \frac{1}{x} )}{x} - \int \frac{sin ( \frac{1}{x} ) - 2 x cos ( \frac{1}{x} )}{x ^{3}} d x]_{0}^{1}

\int \frac{sin ( \frac{1}{x} ) - 2 x cos ( \frac{1}{x} )}{x ^{3}} d x = \frac{cos ( \frac{1}{x} )}{x} + sin (\frac{1}{x})

= [\frac{cos ( \frac{1}{x} )}{x} - (\frac{cos ( \frac{1}{x} )}{x} + sin (\frac{1}{x}))]_{0}^{1}

Simplify

= [- sin (\frac{1}{x})]_{0}^{1}

Compute the boundaries:

diverges

= diverges

What is the integral from 0 to 1 of (cos(1/x))/(x^2) ?
The integral from 0 to 1 of (cos(1/x))/(x^2) is diverges