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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.
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Example 1 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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There were two points to this last example. The first point was to get you thinking about
how to do these problems. It is important
initially to remember that we are really just asking what we differentiated to
get the given function.
The other point is to recognize that there are actually an
infinite number of functions that we could use and they will all differ by a constant.
Now that we’ve worked an example let’s get some of the
definitions and terminology out of the way.
Definitions
Note that often we will just say integral instead of
indefinite integral (or definite integral for that matter when we get to
those). It will be clear from the
context of the problem that we are talking about an indefinite integral (or
definite integral).
The process of finding the indefinite integral is called integration or integrating f(x). If we need to be
specific about the integration variable we will say that we are integrating f(x) with respect to x.
Let’s rework the first problem in light of the new
terminology.
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Example 2 Evaluate
the following indefinite integral.
Solution
Since this is really asking for the most general
anti-derivative we just need to reuse the final answer from the first
example.
The indefinite integral is,
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A couple of warnings are now in order. One of the more common mistakes that students
make with integrals (both indefinite and definite) is to drop the dx at the end of the integral. This is required! Think of the integral sign and the dx as a set of parenthesis. You already know and are probably quite
comfortable with the idea that every time you open a parenthesis you must close
it. With integrals, think of the
integral sign as an “open parenthesis” and the dx as a “close parenthesis”.
If you drop the dx
it won’t be clear where the integrand ends.
Consider the following variations of the above example.
You only integrate what is between the integral sign and the
dx.
Each of the above integrals end in a different place and so we get
different answers because we integrate a different number of terms each
time. In the second integral the “-9” is
outside the integral and so is left alone and not integrated. Likewise, in the third integral the “ ” is outside the integral and so is
left alone.
Knowing which terms to integrate is not the only reason for
writing the dx down. In the Substitution Rule section we will
actually be working with the dx in
the problem and if we aren’t in the habit of writing it down it will be easy to
forget about it and then we will get the wrong answer at that stage.
The moral of this is to make sure and put in the dx!
At this stage it may seem like a silly thing to do, but it just needs to
be there, if for no other reason than knowing where the integral stops.
On a side note, the dx
notation should seem a little familiar to you.
We saw things like this a couple of sections ago. We called the dx a differential in that section
and yes that is exactly what it is. The dx that ends the integral is nothing
more than a differential.
The next topic that we should discuss here is the
integration variable used in the integral.
Actually there isn’t really a lot to discuss here other than to note
that the integration variable doesn’t really matter. For instance,
Changing the integration variable in the integral simply
changes the variable in the answer. It
is important to notice however that when we change the integration variable in
the integral we also changed the differential (dx, dt, or dw) to match the new variable. This is more important that we might realize
at this point.
Another use of the differential at the end of integral is to
tell us what variable we are integrating with respect to. At this stage that may seem unimportant since
most of the integrals that we’re going to be working with here will only
involve a single variable. However, if
you are on a degree track that will take you into multi-variable calculus this
will be very important at that stage since there will be more than one variable
in the problem. You need to get into the
habit of writing the correct differential at the end of the integral so when it
becomes important in those classes you will already be in the habit of writing
it down.
To see why this is important take a look at the following
two integrals.
The first integral is simple enough.
The second integral is also fairly simple, but we need to be
careful. The dx tells us that we are integrating x’s. That means that we only
integrate x’s that are in the
integrand and all other variables in the integrand are considered to be
constants. The second integral is then,
So, it may seem silly to always put in the dx, but it is a vital bit of notation
that can cause us to get the incorrect answer if we neglect to put it in.
Now, there are some important properties of integrals that
we should take a look at.
Properties of the
Indefinite Integral
Notice that when we worked the first example above we used
the first and third property in the discussion.
We integrated each term individually, put any constants back in and then
put everything back together with the appropriate sign.
Not listed in the properties above were integrals of
products and quotients. The reason for
this is simple. Just like with
derivatives each of the following will NOT work.
With derivatives we had a product rule and a quotient rule
to deal with these cases. However, with
integrals there are no such rules. When
faced with a product and quotient in an integral we will have a variety of ways
of dealing with it depending on just what the integrand is.
There is one final topic to be discussed briefly in this
section. On occasion we will be given and will ask what was. We
can now answer this question easily with an indefinite integral.
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Example 3 If
what was ?
Solution
By this point in this section this is a simple question to
answer.
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In this section we kept evaluating the same indefinite
integral in all of our examples. The
point of this section was not to do indefinite integrals, but instead to get us
familiar with the notation and some of the basic ideas and properties of
indefinite integrals. The next couple of
sections are devoted to actually evaluating indefinite integrals.