information:extraction
Integral representation of the natural logarithm

The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (w = x / t). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor t and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function f (x) = 1/ x again. Therefore, the left hand blue area, which is the integral of f (x) from t to tu is the same as the integral from 1 to u. This justifies the equality (2) with a more geometric proof.