\small\color{grey}\text{Use the ``function defined by an integral" part of the FTC}\\[6pt] \text{(and explicitly \textit{\textbf{not}} the ``evaluation" part of the FTC)}\\[3pt] \text{to find the values of } a \text{ for which} \int_{a}^{x}\cos t\,dt=\sin x.\\[14pt] \text{Then find an antiderivative of } \cos x \text{ which will not equal} \\[6pt]  \int_{a}^{x}\cos t\,dt \hspace{}0.05in\text{ for any value of } a.

\small\color{grey}\\[6pt] \text{If } f  \text{ is continuous on } [a,b] \text{, and } F \text{ is the function defined by} \\[6pt] F(x)=\int_{a}^{b}f(t) \, dt, \hspace{0.2in} a\leq x\leq b\\[6pt] \text{then } F \text{ is continuous on }[a,b] \text{ and differentiable on } (a,b), \\[6pt] \text{and } F \text{ is an antiderivative of } f. \\[6pt]