Suppose we want to evaluate some integral
If this is a difficult integral to evaluate, we can often simplify by subsituting some function x(u) for part of the function f(u). For example, if f(u) = sin(2u), we can let x(u) = 2u, resulting in f(u) = sin(x(u)).
This gives the integral ∫abf(x(u))du. There are two adjustments we need to make. Firstly, since u goes from a to b, x(u) must go from x(a) to x(b). Secondly, in order to integrate with respect to x, we need some way to replace du with dx. We can do this by multiplying du by dx/du and dividing the rest of the integrand by dx/du (this entire operation is equivalent to multiplying by 1).
The resulting formula for change of variables is