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Find a function and constant such that a given integral equation holds

Find a function f and a constant c such that

    \[ \int_c^x f(t) \, dt = \cos x - \frac{1}{2} \qquad \text{for all } x \in \mathbb{R}. \]


Let f(t) = -\sin t and c = \frac{\pi}{3}. Then,

    \begin{align*}  \int_c^x f(t) \, dt &= \int_{\frac{\pi}{3}}^x (- \sin t) \, dt \\  &= \cos t \bigr \rvert_{\frac{\pi}{3}}^x \\  &= \cos x - \cos \left( \frac{\pi}{3} \right) \\  &= \cos x - \frac{1}{2}. \end{align*}

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