Change of Variables
Text The use of substitution in integration requires an application of the Change of Variables Theorem.
Suppose we want to evaluate some integral
∫_{a}_{b}f(u)du.
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 ∫_{a}_{b}f(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
∫_{a}^{b}f(u)du = ∫_{x(a)}^{x(b)}[f(x(u))/x'(u)]dx .
(Note that most introductory calculus texts will write ∫_{a}^{b}f(x)dx = ∫_{u(a)}^{u(b)}[f(u(x))/u'(x)]du (hence the term "U subsitution"). The roles of x and u are switched here to make this section parallel to the two and three variable calculus sections.)
Demos
Change of Variables
 
In this demo, f(u) is expressed as a function of x(u) (i.e. the form in which you enter f(u) here is what the function will look like after substitution). One window displays the original integral as the area under a curve. The second window displays the integral after substitution. Though the graphs are shaped differently, the area under each will always be equal.

Exercises 1. Observe what happens for f(u) = sin(x(u)), x(u) = 1/u, u = 0.5 to 1. The two integrals shown appear to be opposites of each other, but are in fact equal. Explain why this is so.
2. Write a simplified change of variables theorem for the special case that x(u) = a * u for some constant a. Try a few examples in the demo and describe the geometric significance of this type of substitution.
