Evaluating the Integral

The question asks to find the expression whose general integral is $X^4 + C$. We need to evaluate the given options to see which one results in this form.

Option D Analysis: $4\int x^3dx$

1. First, evaluate the indefinite integral $\int x^3dx$. Using the power rule for integration, which states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$), we get:

   $ \int x^3dx = \frac{x^{3+1}}{3+1} + C_1 = \frac{x^4}{4} + C_1 $

   Here, $C_1$ represents the constant of integration.

2. Now, multiply the result by 4, as given in Option D:

   $ 4 \times \left( \frac{x^4}{4} + C_1 \right) $

3. Distribute the 4:

   $ 4 \times \frac{x^4}{4} + 4 \times C_1 = x^4 + 4C_1 $

4. Since $C_1$ is an arbitrary constant, $4C_1$ is also an arbitrary constant. Let $C = 4C_1$. The expression simplifies to:

   $ x^4 + C $

   Assuming $X$ in the question refers to the variable $x$, this matches the general integral $X^4 + C$.

Conclusion

The expression $4\int x^3dx$ evaluates to the general integral $X^4 + C$. Therefore, Option D is the correct answer.