Question

Find the most general antiderivative of the function f(x) = 2x³- 2\3*x² + 9x . Check your answer by differentiation and use \( C \) for the constant of the antiderivative.

A. 1/2*x⁴ - 2/9*x³ + 9/2*x² + C
B. 2/9*x⁴ - 2/9*x³ + 9/2*x² + C
C. 1/2*x⁴ - 1/3*x³ + 9/2*x² + C
D. 2/9*x⁴ - 1/3*x³ + 9/2*x² + C

Answer 1

The most general antiderivative of the function f(x) = 2x³ - 2/3x² + 9x is 1/2x⁴ - 2/9x³ + 9/2x² + C, where C is the constant of integration.

Step-by-step explanation:

To find the most general antiderivative, we need to find the antiderivative of each term separately. The antiderivative of 2x³ is 1/2x⁴, the antiderivative of -2/3x² is -2/9x³, and the antiderivative of 9x is 9/2x². Putting all the terms together, the most general antiderivative is 1/2x⁴ - 2/9x³ + 9/2x² + C, where C is the constant of integration.