Final answer:
The antiderivative of 6x² is (2/3)x³ + C, where C is the constant of integration.
Step-by-step explanation:
To find the antiderivative of 6x², we use the power rule for integration, which states that ∫x^n dx = (1/(n+1))x^(n+1) + C. Applying this rule to the given function, where n is 2, we get:
∫6x² dx = (1/(2+1)) * 6x^(2+1) + C
= (2/3)x³ + C
Here, the 2/3 comes from the division by the new exponent (2+1) and the constant of integration, C, accounts for any constant term that may have been present in the original function but disappears when differentiating.
It's crucial to include the constant of integration because when we differentiate the antiderivative to check our result, any constant term becomes zero. Thus, the final antiderivative of 6x² is (2/3)x³ + C.