How do you evaluate the integral of x^2 + 49 with respect to x, and what role does the constant of integration ‘C’ play in this process?

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To evaluate the integral of the function x2 + 49 with respect to x, we will perform the following steps:

  1. Set up the integral: We can express the integral as follows: 
          ∫ (x² + 49) dx
  2. Integrate each term separately: The integral of a sum can be separated into the sum of the integrals. Thus, we can write: 
          ∫ (x² + 49) dx = ∫ x² dx + ∫ 49 dx
  3. Apply the power rule for integration: The power rule states that the integral of x^n is x^(n+1)/(n+1), where n ≠ -1. The integral of a constant a is simply ax. Therefore: 
          ∫ x² dx = (x³ / 3) + C₁
      ∫ 49 dx = 49x + C₂
  4. Combine the results: Now, we can combine the results of the two integrals: 
          ∫ (x² + 49) dx = (x³ / 3) + 49x + C

    Here, C is the combined constant of integration, which encapsulates both C₁ and C₂.

Thus, the final result of the integral is:

  ∫ (x² + 49) dx = (x³ / 3) + 49x + C

In conclusion, the constant of integration C is essential as it accounts for the indefinite nature of integration; it represents an infinite set of possible constants that could be added to the solution without changing the derivative of the function.

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