Deduce General Formula Using Integration by Parts

by Hugo
(Mexico)

This is a very good exercise for practicing using a recurrence formula and also shows the generality of integration by parts. The problem is to find the integral:

Where k is a non-zero constant, and n a natural number.

If we use integration by parts, putting:

(Decision based on LIATE rule). We get:

Now we note that the integral in the right side of the equation is very similar to the one in the left side. So, a convenient way of visualizing this problem is to define the function:

So, using this notation, the previous equation becomes:

This is what is called a recurrence formula. To see where this is going, let's start with n=0:

Now, using our recurrence formula we can find f(1):

We can further calculate f(2) and f(3) to have a good taste of what the general foruma is:

From these we can deduce a general expression for any n. We have to divide in cases, when n is even and when n is odd. When n is odd, as in f(3), the
last term would be negative, and when n is even, as in f(2), the last term would be positive.

So, our final formula is: