Single-variable calculus from first principles
Substitution (often called u-substitution) is the integration technique that reverses the chain rule. When an integral contains a function and a copy of its derivative, you can collapse the messy composition into a clean, simple integral by renaming the inside.
The recipe: spot an inner function, call it u = g(x), compute du = g′(x) dx, and rewrite the integral entirely in terms of u. If you picked u well, the g′(x) dx piece is already sitting there to become du, and the integral becomes trivial.
Substitution is like changing money into a simpler currency to do a sum, then changing back. The integral is awkward in its original 'currency' of x, so you swap to a clean unit u, do the easy arithmetic there, and convert the answer back to x at the end. Pick the exchange wisely and the messy sum turns into one you can do in your head.