Integrating Using Substitution

Many integrals look ugly only because they're hiding a chain rule in reverse. If you see a function tucked inside another function, and the derivative of that inner piece is also sitting in the integral as a factor, that's the signal. Rename the inner function u, and its derivative becomes du. The messy integral in x turns into a clean integral in u, usually a basic power or trig form you already know how to antidifferentiate.

Mechanically, pick u as the inner function, compute du = u'(x) dx, and solve for dx or match it directly to the leftover factor in the integral. Rewrite every piece of the original integral in terms of u, integrate normally, then substitute u back in terms of x at the end. For definite integrals, you can instead convert the limits of integration into u-values and skip switching back, since you never return to x.

This is the reverse-engineering tool for the chain rule, the same way the chain rule itself is the forward tool for differentiating composite functions. It connects directly to the accumulation ideas running through the unit: once you can find an antiderivative, the Fundamental Theorem of Calculus turns that antiderivative into a number for a definite integral. Substitution shows up constantly later too, since area, volume, and other accumulation problems often produce integrands built from composite functions needing exactly this trick.