We explore more difficult problems involving substitution.

We begin by restating the substitution formula.

We will work out a few more examples.

The next example requires a new technique.

Sometimes it is not obvious how a fraction could have been obtained using the chain rule. A common trick though is to substitute for the denominator of a fraction. Like all tricks, this technique does not always work. Regardless the next two examples present how this technique can be used.

Notice that when . So in a very contrived way, we have just proved

Notice the variable in this next example.

We have just proved

Note that in Example example tan, we could have instead made the substitution This would have gotten us to the answer quicker and without using Example key example. You are encouraged to work this out on your own right now!

We end this section with two more difficult examples.

Again, in the previous example we could have instead made the substitution and avoided using Example key example. In general, any time that you make two successive substitutions in a problem, you could have instead just made one substitution. This one substitution is the composition of the two original substitutions. But sometimes it may not be obvious to make one clever substitution, and so two substitutions makes more sense. The next example helps to demonstrate this.