Integrating Using Substitution

Every time you differentiate a composite function with the chain rule, you get an outer derivative times an inner derivative. Substitution runs that process backward. If an integrand looks like f(g(x)) times g'(x), the piece g'(x) dx is exactly what you need to rewrite the whole thing in terms of a new variable u = g(x). Once that swap happens, the integral turns into the integral of f(u) du, something plain and ordinary, often a basic power rule or integral of cos(u) or 1/u.

Mechanically: set u equal to the inner function, compute du = g'(x) dx, and replace every piece of the original integral, x's and dx included, with u's and du. Integrate in terms of u, then substitute g(x) back in for u at the end so the answer is in terms of x again. If an initial condition is given, like F(0) = 0, plug in after undoing the substitution to solve for the constant C.

This topic is the reverse-engineering half of the chain rule, the same way the rest of Unit 6 builds integration as the reverse of differentiation. It is also the gateway technique for the unit: once a student can recognize a disguised derivative, more advanced methods later in the course, like integration by parts, become about handling integrands where no clean du is sitting there waiting to be found.