考查微积分基本定理与微分运算。A：由不定积分性质，$\frac{d}{dx}\int f(x)\,dx = \frac{d}{dx}[F(x)+C] = f(x)$，正确；B：$d\left[\int f(x)\,dx\right] = d[F(x)+C] = F'(x)dx = f(x)dx$，右侧缺 $dx$，错误；C：$\frac{d}{dx}\int f'(x)\,dx = \frac{d}{dx}[f(x)+C] = f'(x)$，而非 $f(x)$，错误；D：$\int f'(x)\,dx = f(x)+C$，缺少常数 $C$，错误。故选 A。