u-substitution

U-substitution is the first real technique for tackling integrals that don't match the basic patterns. The idea is to rename a messy inner part of the integrand as a single new letter, u, so the whole expression collapses into something simple you already know how to integrate. It is like temporarily putting a complicated phrase in a box, doing the easy work on the box, then unboxing at the end.

It is the chain rule run in reverse. The chain rule says the derivative of a composite is the outer derivative times the inner derivative; integrals built that way carry a hidden factor of the inner function's derivative. So you set u = g(x), compute du = g'(x) dx, and replace both pieces: integral f(g(x)) g'(x) dx becomes integral f(u) du. For example, in integral 2x cos(x^2) dx, let u = x^2, so du = 2x dx, and the integral turns into integral cos(u) du = sin(u) + C = sin(x^2) + C.

The craft is spotting the inner function whose derivative is also sitting (perhaps up to a constant) inside the integral. For definite integrals you have two clean options: change the limits to u-values and finish in u, or convert back to x before plugging in the original limits — but never mix the two by using x-limits with a u-expression. When the needed g'(x) factor is missing or extra, substitution alone won't work and you reach for another method such as integration by parts.