uSubstitution
Sometimes, it is impossible to find the integral of a function using the methods already
discussed. The method of usubstitution allows for the integration of more complex
functions, and is essentially a form of the chainrule in reverse.
The general idea is to change a complex integral involving the variable x and dx into
a simpler integral involving the variable u and du. The simpler integral can be
evaluated, and then the relation between u and x can be used to express the result in
terms of x.
Example: Find (4x  2)^{3}dx
This indefinite integral cannot be evaluated directly by the methods already presented. The first step of a usubstitution
is to pick an appropriate part of the original function to be equal to u. This step is often a matter of trial and
error. In this case, however, the choice seems obvious. Let u = 4x  2. Now, to find the relationship between du and dx,
take the derivative of u with respect to x:
= 4
Rearranging, this means that du = 4dx, or dx = du.
With this information, we can substitute u and du into the original integral:
(4x  2)^{3}dx = (u)^{3}()du 

To evaluate this integral, the constant can first be pulled out:
=u^{3}du. By basic integration rules, this is equal to u^{4}. Now, this result should be
translated back into a function of the variable x by using the relation u = 4x  2. Substituting x back into the formula
yields:
(4x  2)^{4}