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

by RoRi

October 15, 2015

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,

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

Stumbling Robot

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

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