We explore more difficult problems involving substitution.
We will work out a few more examples.
The next example requires a new technique.
Now we consider the integral we are trying to compute and we substitute using our work above. Write with me
However, we cannot continue until each is replaced. We know that
so now we may replace At this point, we are close to being done. Write
Now recall that . Hence our final answer is
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.
But this cancels perfectly with the numerator! So we have that
Notice that when . So in a very contrived way, we have just proved
Notice the variable in this next example.
We then make the substitution and so
But this is the same problem as Example key example! And so we know that
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.
But now we are back to Example key example, and so we know that
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.
From here we now make the second (and more obvious) substitution Then , and