please help and show all work.?

Question

What happens when you apply the power rule for integration to the function 1/x?

Answer 1

In general, the power rule for integration says:
∫(x^a) dx = [x^(a+1)]/(a+1) + C, but if you apply this to 1/x, it gives that ∫[x^(-1)] dx = [x^(0)]/0 +C, which is clearly undefined. However, we know that d/dx[ln(|x|)]=1/x, and so ∫1/x dx = ln(|x|) + C by definition.

Answer 2

The rule does not applied to n= -1
the answer is ln|x| + c