One pattern unlocks the logarithmic rule: a function in the denominator with its own derivative on top. Then the integral of f'(x)/f(x) is ln|f(x)| + C. This covers the n = -1 case the power rule cannot reach, and the absolute-value bars let a single formula work on both branches of the curve.
The logarithmic rule is one of the most useful rules in integration, and the hardest part of it is recognising when it applies. There is one pattern to look for: a function in the denominator with its own derivative sitting in the numerator.
Whatever sits in the denominator goes straight inside the logarithm, wrapped in absolute value bars, and the constant is added as always. Nothing is expanded, substituted or simplified first.
See a function and its derivative stacked like that, and reach for immediately.
is not a different kind of logarithm. It is the ordinary logarithm with the base fixed at the number , and the shorter name simply saves writing the base every time.
Why in particular deserves its own notation is a question for another lesson. Here it is enough to read as "the exponent that turns into this number".
In each case, name the denominator, differentiate it, and check that the result is what is sitting on top.
The constant in the second example differentiates to zero, which is why it leaves no trace in the numerator. Keeping the bars costs nothing even when the denominator can never be negative, as with .
The power rule adds one to the exponent and divides by the new exponent. For it breaks:
Division by zero, so the rule simply does not reach this case. But is , and that is the logarithmic pattern in its simplest form: the denominator is , and its derivative is .
The one exponent the power rule cannot handle is exactly the one the logarithmic rule was waiting for.
A logarithm is undefined for negative inputs. Yet exists on the left of the axis too, and there is a perfectly real region between that branch and the -axis.
On the left the region sits below the axis, which would make it negative, but it is also traversed leaving-wards. Two negatives meet and the result comes out positive, matching what the bars give.