Final answer:
To find an antiderivative f(x) for f'(x) = 10/(21x^2) - 12x^5 with the condition f(1) = 0, integrate f'(x) term by term then use the condition to find the integration constant, resulting in f(x) = -10/(21x) - 2x^6 + 2 + 10/21.
Step-by-step explanation:
The original question is asking to find an antiderivative f(x) of the function f'(x) = 10/(21x^2) - 12x^5, with the initial condition that f(1) = 0. To find f(x), we integrate the function f'(x) with respect to x. This involves finding the indefinite integral and then using the initial condition to solve for the constant of integration.
First, we integrate the function term by term:
- For the term 10/(21x^2), the antiderivative is -10/(21x).
- For the term -12x^5, the antiderivative is -2x^6.
Combining these gives us the general antiderivative function:
f(x) = -10/(21x) - 2x^6 + C
Next, we apply the initial condition f(1) = 0 to solve for C:
0 = -10/21 - 2 + C
Solving for C gives us C = 2 + 10/21, and therefore:
f(x) = -10/(21x) - 2x^6 + 2 + 10/21
This is the antiderivative that satisfies the given condition.