Answer :
Final answer:
The most general antiderivative of the function f(x) = 6x^5 - 7x^4 - 6x² is 6x^6 - 7x^5 - 2x^3 + C.
Explanation:
To find the antiderivative of the given function, we integrate each term separately. For each term, we add 1 to the exponent and divide by the new exponent. Thus, the antiderivative of 6x^5 is (6/6)x^6 = x^6, the antiderivative of -7x^4 is (-7/5)x^5, and the antiderivative of -6x² is (-6/3)x^3 = -2x^3. We then combine these antiderivatives and add the constant of integration, C.
Mathematically, we integrate each term:
∫(6x^5 - 7x^4 - 6x²)dx = ∫6x^5 dx - ∫7x^4 dx - ∫6x² dx
= (6/6)x^6 - (7/5)x^5 - (6/3)x^3 + C
= x^6 - (7/5)x^5 - 2x^3 + C
Therefore, the correct option representing the antiderivative is option A) 6x^6 - 7x^5 - 2x^3 + C.
