5. (20 points) Let $f^{\prime \prime}(x)=e^{x}-2 \sin x$ (a) (6pts) Find the general antiderivative of $f^{\prime \prime}(x)$. (b) (6pts) Find specific antiderivative of $f^{\prime \prime}(x)$ if $f^{\prime}(0)=0$. (c) (8pts) Find $f(x)$ if $f(\pi / 3)=0$.

Step 1 ：The question asks for the antiderivative of \(f^{\prime \prime}(x)=e^{x}-2 \sin x\). The antiderivative, also known as the integral, of a function is the reverse of taking the derivative. Therefore, to find the antiderivative of \(f^{\prime \prime}(x)\), we need to find a function whose derivative is \(f^{\prime \prime}(x)\).

Step 2 ：The antiderivative of \(e^{x}\) is \(e^{x}\) and the antiderivative of \(\sin x\) is \(-\cos x\). Therefore, the general antiderivative of \(f^{\prime \prime}(x)\) is \(e^{x} + 2\cos x + C\), where \(C\) is the constant of integration.

Step 3 ：Final Answer: The general antiderivative of \(f^{\prime \prime}(x)\) is \(\boxed{e^{x} + 2\cos x + C}\).