antiderivatives

[tgrrrrr1976]

I am having trouble visualizing the steps I need to go through to find the antiderivatives of some types of problems. I know that 1/x = ln|x| but for some reason, I cannot put together in my head what to do with a problem like 1/(x^2) . I find the answer and think "of course, -1/x differentiates to 1/(x^2)" but I am having a very difficult time carrying out this procedure in reverse. I know this is a vague sort of question, but I was hoping someone might have a tip or two that might help me visualize these better. Or maybe there is a 'trick' I am unaware of?

 

[BigGlenntheHeavy]

\(\displaystyle \int\frac{1}{x^{2}}dx \ = \ \int x^{-2}dx, \ = \ \frac{x^{-1}}{-1} \ = \ -x^{-1} \ = \ \frac{-1}{x}.\)

\(\displaystyle Check: \ D_x\bigg[\frac{-1}{x}\bigg] \ = \ D_x[-x^{-1}] \ = \ x^{-2} \ = \ \frac{1}{x^{2}}\)

\(\displaystyle When \ doing \ integrals \ add, \ when \ doing \ derivatives, \ subtract.\)

 

[tgrrrrr1976]

thanks a lot! I was just over complicating it in my head (as usual). The rule applies just the same, I was just trying to skip too many steps in my head and not recognizing that the end product was just the result of some algebra.