The integral of 3x^2 - 4x + 1 equals x^3 - 2x^2 + x. Is this always correct?

Where is the constant of interation?

wouldn't x^3 - 2x^2 + x + 23 also be correct?

To check whether the given integral is correct, we can differentiate the function x^3 - 2x^2 + x and see if we obtain the original function 3x^2 - 4x + 1. Let's differentiate x^3 - 2x^2 + x:

d/dx (x^3 - 2x^2 + x) = 3x^2 - 4x + 1

We can see that the derivative matches the original function. Therefore, the given integral is indeed correct.

Now, regarding the "constant of integration," in indefinite integrals, we include a constant term while integrating to account for all possible antiderivatives of the integrand. In this case, since we integrated 3x^2 - 4x + 1, the constant of integration is included. However, it's represented by the "+ C" in our answer, which technically accounts for any constant value.

To address your second question, x^3 - 2x^2 + x + 23 is not equivalent to x^3 - 2x^2 + x as the constant term has changed from "+ C" to "+ 23". In this case, the "+ 23" represents a specific constant value and is not part of the general '+ C' term.