Integration by substitution in Real Analysis is a fundamental theorem stating that if $u(x)$ is a continuously differentiable function and its derivative is continuous on the interval of integration, then the integral with respect to $x$ can be transformed into an equivalent integral with respect to $u$, provided the limits are adjusted accordingly. This method relies strictly on the Chain Rule for indefinite integrals and extends formally through Fundamental Theorem of Calculus Part 1 under smoothness conditions regarding composition functions. It serves as a canonical change-of-variables technique within Riemann integration theory, bridging algebraic manipulation with geometric area preservation in multivariable domains without invoking measure-theoretic complexity.
This concept covers two equivalent methods for evaluating a definite integral via u-substitution: (1) computing the indefinite antiderivative through substitution, reverting to the original variable,…