Final answer:
The antiderivative of the polynomial function 6x⁵ − 7x⁴ − 9x² is found by using the power rule for integration to obtain ⅓x⁶ − ⅔x⁵ − 3x³ + C, where C is the constant of integration.
Step-by-step explanation:
Finding an Antiderivative
The student has been tasked with finding the antiderivative of the polynomial function 6x⁵ − 7x⁴ − 9x². An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function.
To find the antiderivative we will apply the power rule for integration, which states that the antiderivative of x raised to the power of n, where n is a real number other than -1, is x raised to the power of n+1 divided by n+1 plus a constant of integration C.
To find the antiderivative of 6x⁵ − 7x⁴ − 9x², we will apply the power rule to each term separately:Combining all the terms, the final antiderivative of the original function is ⅓x⁶ − ⅔x⁵ − 3x³ + C, where C is the constant of integration. It is important to include this constant because the derivative of any constant is zero, which means adding a constant does not affect the derivative of a function.