Final answer:
The most general antiderivative of the function 8x^4 - 6x^-4 - 3 is obtained by integrating each term separately to get 8/5x^5 + 2x^-3 - 3x + C, where C is the constant of integration.
Step-by-step explanation:
The student is asking about finding the most general antiderivative of the function 8x4 - 6x-4 - 3. To find the most general antiderivative, also known as the indefinite integral, we integrate each term of the function separately and add a constant of integration, C, at the end.
For the term 8x4, the antiderivative is 8/5x5 because when we take the derivative of x5, we get 5x4, and to compensate for the 5, we multiply by 8/5.
For -6x-4, which can also be written as -6/x4, the antiderivative is 2/x3 or 2x-3, since the derivative of x-3 gives us -3x-4, and we multiply by -2 to get the coefficient -6.
The constant term -3 has a straightforward antiderivative of -3x. Therefore, the most general antiderivative of the function is 8/5x5 + 2x-3 - 3x + C, where C is the constant of integration.