Question

step 1
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 - 7 x^{3}+5 x^{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
\begin{aligned}g(x) &=\frac{6 - 7 x^{3}+5 x^{6}}{x^{6}} \\&=\frac{6}{x^{6}}-\frac{7 x^{3}}{x^{6}}+\frac{5 x^{6}}{x^{6}} \\&=6 x^{-6}-7 x^{-3}+5end{aligned}
step 2
now we can say the most general antiderivative of ( g(x)=\frac{6 - 7 x^{3}+5 x^{6}}{x^{6}} ) is
g(x)=6left(\frac{1}{-5} x^{-5}
ight)-7left(\frac{1}{-2} x^{-2}
ight)+5 x + c
step 3
simplifying and remembering to use ( c ) for the constant of the antiderivative we have
g(x)=square

Explanation:

Step1: Recall the power - rule for integration

The power - rule for integration is \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\), where \(n
eq - 1\).

For the function \(g(x)=6x^{-6}-7x^{-3}+5\), we integrate each term separately.

For the first term \(\int6x^{-6}dx\):
Using the power - rule with \(n=-6\), we have \(6\times\frac{x^{-6 + 1}}{-6 + 1}=\frac{6x^{-5}}{-5}=-\frac{6}{5x^{5}}\)

Step2: Integrate the second term

For the second term \(\int-7x^{-3}dx\):
Using the power - rule with \(n =-3\), we get \(-7\times\frac{x^{-3+1}}{-3 + 1}=-7\times\frac{x^{-2}}{-2}=\frac{7}{2x^{2}}\)

Step3: Integrate the third term

For the third term \(\int5dx\):
Since \(\int a dx=ax + C\) (where \(a = 5\)), we have \(5x\)

Answer:

\(G(x)=-\frac{6}{5x^{5}}+\frac{7}{2x^{2}}+5x + C\)