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Overview of Integral U-Substitutions in One Dimension
Introduction: where does u-substitution belong in the curriculum?
The change of variables, otherwise known as u-substitution, might be one of the most dreaded topics in calculus, but in my opinion, it’s also one of the most fun. While change of variables techniques extend from relatively simple examples in one dimension to higher-dimensional examples involving matrices, I’m going to stick to examples in one dimension for this article.
In the high school AP curriculum in the United States, this material falls within AP Calculus AB. In college, this material will be covered in Calculus 1 or Calculus 2, depending on your university’s curriculum.
The basic idea behind u-substitution is that you are transforming the integrand, so that instead of integrating with respect to one variable and domain, you are integrating with respect to a different one. This relationship is most apparent when solving definite integral problems, though u-substitution is perfectly viable for solving indefinite integrals as well. Here is a more symbolic way to consider how u-substitution works at a high level.
