Question

it is not true that if f and g are antiderivatives of f and g, respectively, then f/g is an antiderivative of f/g. therefore, in order to find the antiderivative of ( g(x)=\frac{6 - 7x^{3}+5x^{6}}{x^{6}} ), we must use the properties that ( \frac{1}{x^{n}}=x^{-n} ) and ( \frac{x^{p}}{x^{n}}=x^{p - n} ) to rewrite the fraction as:

( g(x)=\frac{6 - 7x^{3}+5x^{6}}{x^{6}} )

( =\frac{square}{x^{6}}-\frac{7x^{3}}{x^{6}}+\frac{5x^{6}}{x^{6}} )

( =square x^{-6}-7x^{-3}+square ).

Explanation:

Step1: Split the fraction

Using the rule \(\frac{a + b+ c}{d}=\frac{a}{d}+\frac{b}{d}+\frac{c}{d}\), we split \(\frac{6 - 7x^{3}+5x^{6}}{x^{6}}\) into \(\frac{6}{x^{6}}-\frac{7x^{3}}{x^{6}}+\frac{5x^{6}}{x^{6}}\).

Step2: Use the power - rule for exponents

Recall that \(\frac{x^{m}}{x^{n}}=x^{m - n}\). For \(\frac{6}{x^{6}}\), since \(6 = 6x^{0}\), then \(\frac{6x^{0}}{x^{6}}=6x^{0-6}=6x^{-6}\). For \(\frac{5x^{6}}{x^{6}}\), using \(\frac{x^{m}}{x^{n}}=x^{m - n}\), we have \(5x^{6 - 6}=5x^{0}=5\).

Answer:

The first blank is \(6\), the second blank is \(6\), and the third blank is \(5\).