find an antiderivative of (9 x^2 + 9 x^4)/x^5 in the variable x where x ≠ 0. remember to include a \+ c\ if…

Explanation:

Step1: Simplify the function

First, rewrite $\frac{9x^{2}+9x^{4}}{x^{5}}$ as $\frac{9x^{2}}{x^{5}}+\frac{9x^{4}}{x^{5}} = 9x^{-3}+9x^{-1}$.

Step2: Integrate term - by - term

The antiderivative of $9x^{-3}$ using the power rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$) is $9\times\frac{x^{-3 + 1}}{-3+1}=-\frac{9}{2}x^{-2}$. The antiderivative of $9x^{-1}$ is $9\int\frac{1}{x}dx=9\ln|x|$.

Step3: Combine results

The antiderivative of $9x^{-3}+9x^{-1}$ is $-\frac{9}{2}x^{-2}+9\ln|x|+C$.

Answer:

$-\frac{9}{2}x^{-2}+9\ln|x|+C$