question\nfind the most general form of the antiderivative, $f(x)$, of the function $f(x)=\frac{x…

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{x - 4}{x^{9}}$ as $f(x)=\frac{x}{x^{9}}-\frac{4}{x^{9}}=x^{- 8}-4x^{-9}$.

Step2: Use the power - rule for antiderivatives

The power - rule for antiderivatives is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int x^{-8}dx$, we have $\frac{x^{-8 + 1}}{-8 + 1}=\frac{x^{-7}}{-7}=-\frac{1}{7x^{7}}$. For $\int-4x^{-9}dx=-4\int x^{-9}dx=-4\times\frac{x^{-9 + 1}}{-9 + 1}=-4\times\frac{x^{-8}}{-8}=\frac{1}{2x^{8}}$.

Step3: Combine the results and add the constant of integration

$F(x)=-\frac{1}{7x^{7}}+\frac{1}{2x^{8}}+C$.

Answer:

$F(x)=-\frac{1}{7x^{7}}+\frac{1}{2x^{8}}+C$