H2 - Integration Using u Substitution
The method of substitution hinges on the following formula in which u stands for a
differentiable function of x.
To justify this formula, let F be an antiderivative of
, so that
. If u is a differentiable function of x, then
Example: Evaluate
In practice the substitution process is carried out as follows:
.
Integration by Substitution
Step 1. Make a choice for
, say
.
Step 2. Compute
Step 3. Make the substitution
At this stage, the entire integral must be in terms of
the case, try a different choice for .
; no x's should remain. If this is not
Step 4. Evaluate the resulting integral.
Step 5. Replace u by
, so the final answer is in terms of x.
Example: Evaluate
Another easy u-substitution occurs when the integrand is the derivative of a known
function, except for a constant that multiplies or divides the independant variable. The
following example illustrate such integrals.
Example: Evaluate
Example: Evaluate
As you develop the technique of integration using u substitution, students should have an
integration formula sheet in front of them. Students should view the formulas as moulds
or models. Their challenge is to convert the given integral into one of these moulds by
using appropriate substitution. After students gain experience, they will make more
intelligent guesses for u. One pointer I can share with students is you should not simply
substitute u for x, because you have not changed the original problem. The second peice
of advice I can share is to work lots of problems. The more problems you perform gains
you experience. This experience will pay off later when you start examining different
techniques to solve integrals.