Integration by Substitution (u-Substitution)
The Big Idea
Integration by substitution is the reverse of the Chain Rule for differentiation. Just as the Chain Rule lets us differentiate composite functions, u-substitution lets us integrate composite functions by simplifying them into a form we recognize.
Formal Definition
If $g(x)$ is a differentiable function whose range is contained in the domain of $f$, then:
$$ \int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du $$
where $u = g(x)$ and $du = g'(x) \, dx$.
In essence, we identify an inner function $u = g(x)$, compute its differential $du = g'(x)\,dx$, and rewrite the entire integral in terms of $u$.