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Find the integral from 1 to 2 of x-2 sin (1/x)

by RoRi

Evaluate the following integral:

    \[ \int_1^2 x^{-2} \sin \left( \frac{1}{x} \right) \, dx. \]

We make the substitution,

    \[ u = \frac{1}{x} \qquad \implies \qquad du = -\frac{1}{x^2} \, dx. \]

Giving us new bounds of integration,

    \[ u(1) = 1 \qquad \text{and} \qquad u(2) = \frac{1}{2}. \]

Then we can evaluate

    \begin{align*} \int_1^2 \frac{1}{x^2} \sin \left( \frac{1}{x} \right) \,dx &= -\int_1^{\frac{1}{2}} \sin u \, du \\ &= \int_{\frac{1}{2}}^1 \sin u \, du \\ &= -\cos u \Big \rvert_{\frac{1}{2}}^1 \\ &= \cos \frac{1}{2} - \cos 1. \end{align*}

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