Find the function \(f(x)\) described by the given initial value problem.
\(f'(x) = \sin x, \quad f(0) = 10\)
\(f(x) =\) ?
Integrating \(f'(x)\) with respect to \(x\) yields
\(f(x) = -\cos(x) + C\),
where \(C\) is a constant we can solve for given the initial condition. Because we are given \(f(0) = 10,\) we can say
\(10 = f(0) = -\cos(0) + C = -1 + C\).
This implies that \(C = 11.\) Hence,
\(f(x) = -\cos(x) + 11.\)