Which of the following statements are true? Select all that apply. Each correct response is worth positive fractional marks (to a max of 1 mark) and each incorrect response is worth negative fractional marks (to a min of 0 ). A. If $\frac{d}{d x} h(x)=g(x)$ then $\int h(x) d x=g(x)+C$ for some arbitrary constant $C$. B. $\frac{d}{d x} \int h(x) d x=h(x)$ C. If $\frac{d}{d x} h(x)=f(x)$ then $\int f(x) d x=C h(x)$ for some arbitrary constant $C$. D. $\int h^{\prime}(x) d x=h(x)+C$ for some arbitrary constant $C$. E. If $\frac{d}{d x} h(x)=f(x)$ then $\int f(x) d x=h(x)+C$ for some arbitrary constant $C$.
Solution
Step 1 ：The question is asking about the properties of derivatives and integrals, which are fundamental concepts in calculus.
Step 2 ：Statement A is incorrect. The integral of the derivative of a function is the original function plus a constant, not the derivative itself.
Step 3 ：Statement B is correct. The derivative of the integral of a function is the original function. This is known as the Fundamental Theorem of Calculus.
Step 4 ：Statement C is incorrect. The integral of a function is not equal to a constant times the original function.
Step 5 ：Statement D is correct. The integral of the derivative of a function is the original function plus a constant.
Step 6 ：Statement E is correct. The integral of a function that is the derivative of another function is the original function plus a constant.
Step 7 ：The correct statements are B, D, and E.
