Integrals homework help question

Question

My question is about number 17. It shows the question first then next step it goes to $\frac{1}{x^2}$. Can't I just leave that as is instead of making it $x^{-2}$, take anti derivative which is $\ln (x^2)$?

Answer 1

we have $$-\int x^{-2}dx$$ from $$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$$ if $$n\ne -1$$ we get $$x^{-1}+C$$

Answer 2

The antiderivative of $1/x^2$ is not $\ln x^2$. In general, the antiderivative of $1/\mbox{crud}$ is not $\ln \mbox{crud}.$ Only the antiderivative of $1/x$ is $\ln x$.