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# Find the function f(x) described by the given initial value problem.

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Find the function $$f(x)$$ described by the given initial value problem.

$$f'(x) = \sin x, \quad f(0) = 10$$

$$f(x) =$$ ?

May 3, 2022

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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,

$$f(x) = -\cos(x) + 11.$$

May 3, 2022