Let's solve the integral of the constant 2494. When integrating a constant, we use the fundamental rule that the integral of any constant c with respect to x equals c times x plus the constant of integration C. Applying this rule to our problem, the integral of 2494 is simply 2494x plus C.
Now let's work through this step by step. First, we identify that our function is the constant 2494. Second, we apply the integration rule for constants, which states that the integral of a constant equals the constant times x. Finally, we write our result as 2494x plus the constant of integration C.
The constant of integration C is crucial because it represents all possible antiderivatives. When we differentiate any constant, we get zero, so there are infinitely many functions whose derivative is 2494. These functions differ only by a constant value, which we represent as C. The graph shows three examples of antiderivatives, all parallel lines with the same slope but different vertical positions.
To verify our answer, let's take the derivative of our result. The derivative of 2494x plus C equals the derivative of 2494x plus the derivative of C. Using the power rule, the derivative of 2494x is 2494, and the derivative of any constant is zero. Therefore, we get 2494 plus zero, which equals 2494. This confirms that our integration is correct.
In conclusion, the integral of 2494 is 2494x plus C. This follows the fundamental rule that the integral of any constant c is simply c times x plus the constant of integration. Remember, integration and differentiation are inverse operations, which is why we can verify our answer by taking the derivative and getting back our original function.