Indefinite Integrals
In the past two chapters we�ve been given a function,
 ,
and asking what the derivative of this function was. Starting with this section
we are not going to turn things around. We now want to ask what function we
differentiated to get the function
 .
Let�s take a quick look at an example to get us started.
Example What
function did we differentiate to get the following function.
Solution
Let�s actually start by getting the derivative of this
function to help us see how we�re going to have to approach this problem. The
derivative of this function is,
The point of this was to remind us of how differentiation
works. When differentiating powers of x we multiply the term by the
original exponent and then drop the exponent by one.
Now, let�s go back and work the problem. In fact let�s
just start with the first term. We got x4 by differentiating
a function and since we drop the exponent by one it looks like we must have
differentiated x5. However, if we had differentiated x5
we would have 5x4 and we don�t have a 5 in front our first
term, so the 5 needs to cancel out after we�ve differentiated. It looks then
like we would have to differentiate
 in
order to get x4.
Likewise for the second term, in order to get 3x
after differentiating we would have to differentiate
 .
Again, the fraction is there to cancel out the 2 we pick up in the
differentiation.
The third term is just a constant and we know that if we
differentiate x we get 1. So, it looks like we had to differentiate -9x
to get the last term.
Putting all of this together gives the following function,
Our answer is easy enough to check. Simply differentiate
 .
So, it looks like we got the correct function. Or did we?
We know that the derivative of a constant is zero and so any of the following
will also give
 upon
differentiating.
In fact, any function of the form,
will give
 upon
differentiating.
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