Question

consider the function \\(f(x) = \frac{9}{x^3} - \frac{6}{x^7}\\).

let \\(f(x)\\) be the antiderivative of \\(f(x)\\) with \\(f(1) = 0\\).
then \\(f(x) =\\)

Explanation:

Rewrite the function using negative exponents

$$ f(x) = 9x^{-3} - 6x^{-7} $$

Find the general antiderivative

$$ F(x) = \int (9x^{-3} - 6x^{-7}) \, dx = 9 \cdot \frac{x^{-2}}{-2} - 6 \cdot \frac{x^{-6}}{-6} + C = -\frac{9}{2x^2} + \frac{1}{x^6} + C $$

Solve for the constant C using the initial condition

$$
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Answer:

Consider the function \(f(x) = \frac{9}{x^3} - \frac{6}{x^7}\).

Let \(F(x)\) be the antiderivative of \(f(x)\) with \(F(1) = 0\).
Then \(F(x) =\) <blank>\(-\frac{9}{2x^2} + \frac{1}{x^6} + \frac{7}{2}\)</blank>