Which of the following statements is true?
A. If $f^{\prime}(x)=2$ for all $x$ and if and $f(0)=0$ then $\int f(x) d x=k x+C$ for some constant $C$.
B. If $\int f(x) d x=C$ then $f(x)=C+x$ for some constant $C$.
C. $\int \frac{1}{\ln (x)} d x=\ln |\ln (x)|+C$ for some constant $C$.
D. If $f^{\prime}(x)=2$ for all $x$ and if and $f(0)=0$ then $\int f(x) d x=x^{2}+C$ for some constant $C$.

Step 1 ：The question is asking us to determine which of the given statements is true.

Step 2 ：Let's analyze each option:

Step 3 ：Option A: If $f^{\prime}(x)=2$ for all $x$ and if and $f(0)=0$ then $\int f(x) d x=k x+C$ for some constant $C$. This statement is saying that if the derivative of a function is a constant, then the integral of the function is a linear function. This is true because the integral of a constant is a linear function. However, the constant of integration is not specified in this statement, so we cannot determine if it is true or false without more information.

Step 4 ：Option B: If $\int f(x) d x=C$ then $f(x)=C+x$ for some constant $C$. This statement is saying that if the integral of a function is a constant, then the function is a linear function. This is not necessarily true. The integral of a function can be a constant, but the function itself can be any function whose derivative is zero, not just a linear function.

Step 5 ：Option C: $\int \frac{1}{\ln (x)} d x=\ln |\ln (x)|+C$ for some constant $C$. This statement is saying that the integral of $1/\ln(x)$ is $\ln|\ln(x)|+C$. This is not true. The integral of $1/\ln(x)$ is a special function called the logarithmic integral, denoted as $\text{li}(x)$, not $\ln|\ln(x)|+C$.

Step 6 ：Option D: If $f^{\prime}(x)=2$ for all $x$ and if and $f(0)=0$ then $\int f(x) d x=x^{2}+C$ for some constant $C$. This statement is saying that if the derivative of a function is a constant, then the integral of the function is a quadratic function. This is true because the integral of a constant is a linear function, and the integral of a linear function is a quadratic function. Moreover, the constant of integration is specified in this statement, so we can determine that it is true.

Step 7 ：Based on the analysis, option D seems to be the correct answer. Let's confirm this by calculating the integral of a function whose derivative is a constant.

Step 8 ：The integral of the function is indeed a quadratic function, which confirms that option D is correct.

Step 9 ：Final Answer: The correct statement is \(\boxed{\text{D}}\). If $f^{\prime}(x)=2$ for all $x$ and if and $f(0)=0$ then $\int f(x) d x=x^{2}+C$ for some constant $C$.