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Integrating a polynomial applies the reverse power rule term by term: xⁿ becomes x^(n+1)/(n+1). Every antiderivative differs by a constant, which is why C appears.
Reverse power rule
∫a·xⁿ dx = a·x^(n+1)/(n+1) + C, for n ≠ −1
1.5x⁴ − (4/3)x³ + 3x + C, and differentiating returns the original.
Because the derivative of any constant is zero, so infinitely many functions share the same derivative.