{'fraction_non_alphanumeric': 0.011940298507462687, 'fraction_numerical': 0.0, 'mean_word_length': 5.134146341463414, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
An alternative proof of the integral of a logarithm

The conventional proof of the integral of a logarithm utilises integration by parts, resulting in a justification that is both simple and elegant. Both are introduced to A level mathematics students in the United Kingdom. However, in the American education system, integration by parts is a technique taught in Calculus II after the integral of a logarithm is introduced in Calculus I. Therefore, the integral of a logarithm is often given without rigorous justification. What follows is an alternate proof that employs logarithmic differentiation and basic rules of integration presented early in calculus studies. In addition to providing American students with a comprehensible proof for the integral of a logarithm, the following proof offers students of the UK yet another example of how a number of elementary techniques can be combined to achieve a result that is neither obvious nor intuitive, an invaluable skill to the aspiring mathematician.