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Compute the integral

by RoRi

Compute the following integral:

    \[ \int_0^x \left( \sin 2w + \cos \frac{w}{2} \right) \, dw. \]

We have

    \begin{align*} \int_0^x \left( \sin 2w + \cos \frac{w}{2} \right) \, dw &= \frac{1}{2} \int_0^{2x} \sin w \, dw + 2 \int_0^{\frac{x}{2}} \cos w \, dw \\ &= \frac{1}{2} (-\cos w)\biggr \rvert_0^{2x} + 2 (\sin w) \biggr \rvert_0^{\frac{x}{2}} \\ &= -\frac{1}{2} \cos (2x) + \frac{1}{2} + 2 \sin \frac{x}{2} \\ &= \frac{1}{2} - \frac{1}{2} \cos (2x) + 2 \sin \frac{x}{2}. \end{align*}

On the second line we used the expansion/contraction of the interval of integration twice (one expansion, one contraction).

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