Consider the function H defined by
H(x)=∫(π3)(5x4−x2+4e3x)dx.
Throughout this problem, treat π and 3 as exact constants.
(a) Calculate H(x) by finding an explicit antiderivative of the integrand. Present your result in its simplest algebraic form and include the constant of integration C.
(b) Differentiate the expression you obtained in part (a) to confirm that it reproduces the original integrand. Show all relevant derivative calculations.
(c) Suppose that H(1)=8. Determine the particular value of the integration constant C that satisfies this condition, and write the finalized expression for H(x). Clearly indicate all steps in your reasoning.