A calculus student is working with the indefinite integral
∫3x2(x3+4)5dx
and proposes the antiderivative
F(x)=6(x3+4)6+C.
Use differentiation as a means of confirmation.
(a) Compute F′(x) directly from the given F(x). Show every step of your differentiation.
(b) Compare F′(x) to the original integrand 3x2(x3+4)5. Determine whether the proposed F(x) is a correct antiderivative, and justify your conclusion.
(c) If the expression in part (b) does not exactly reproduce the integrand, provide a corrected antiderivative F(x) and give a brief explanation of how your verification led you to the correction.