In this section we need to address a couple of topics about
the constant of integration. Throughout
most calculus classes we play pretty fast and loose with it and because of that
many students don’t really understand it or how it can be important.
First, let’s address how we play fast and loose with
it. Recall that technically when we
integrate a sum or difference we are actually doing multiple integrals. For instance,
Upon evaluating each of these integrals we should get a
constant of integration for each integral since we really are doing two
integrals.
Since there is no reason to think that the constants of
integration will be the same from each integral we use different constants for
each integral.
Now, both c and k are unknown constants and so the sum
of two unknown constants is just an unknown constant and we acknowledge that by
simply writing the sum as a c.
So, the integral is then,
We also tend to play fast and loose with constants of
integration in some substitution rule problems.
Consider the following problem,
Technically when we integrate we should get,
Since the whole integral is multiplied by ,
the whole answer, including the constant of integration, should be multiplied
by . Upon multiplying the through the answer we get,
However, since the constant of integration is an unknown
constant dividing it by 2 isn’t going to change that fact so we tend to just
write the fraction as a c.
In general, we don’t really need to worry about how we’ve
played fast and loose with the constant of integration in either of the two
examples above.
The real problem however is that because we play fast and
loose with these constants of integration most students don’t really have a
good grasp oF them and don’t understand that there are times where the
constants of integration are important and that we need to be careful with
them.
To see how a lack of understanding about the constant of
integration can cause problems consider the following integral.
This is a really simple integral. However, there are two ways (both simple) to
integrate it and that is where the problem arises.
The first integration method is to just break up the
fraction and do the integral.
The second way is to use the following substitution.
Can you see the problem?
We integrated the same function and got very different answers. This doesn’t make any sense. Integrating the same function should give us
the same answer. We only used different
methods to do the integral and both are perfectly legitimate integration
methods. So, how can using different
methods produce different answer?
The first thing that we should notice is that because we
used a different method for each there is no reason to think that the constant
of integration will in fact be the same number and so we really should use
different letters for each.
More appropriate answers would be,
Now, let’s take another look at the second answer. Using a property of logarithms we can write
the answer to the second integral as follows,
Upon doing this we can see that the answers really aren’t
that different after all. In fact they
only differ by a constant and we can even find a relationship between c and k. It looks like,
So, without a proper understanding of the constant of
integration, in particular using different integration techniques on the same
integral will likely produce a different constant of integration, we might
never figure out why we got “different” answers for the integral.
Note as well that getting answers that differ by a constant
doesn’t violate any principles of calculus.
In fact, we’ve actually seen a fact that suggested that this might
happen. We saw a fact in the Mean Value Theorem section that said that
if then . In other words, if two functions have the
same derivative then they can differ by no more than a constant.
This is exactly what we’ve got here. The two functions,
have exactly the same derivative,
and as we’ve shown they really only differ by a constant.
There is another integral that also exhibits this
behavior. Consider,
There are actually three different methods for doing this
integral.
Method 1 :
This method uses a trig formula,
Using this formula (and a quick substitution) the integral
becomes,
Method 2 :
This method uses the substitution,
Method 3 :
Here is another substitution that could be done here as
well.
So, we’ve got three different answers each with a different
constant of integration. However,
according to the fact above these three answers should only differ by a
constant since they all have the same derivative.
In fact they do only differ by a constant. We’ll need the following trig formulas to
prove this.
Start with the answer from the first method and use the
double angle formula above.
Now, from the second identity above we have,
so, plug this in,
This is then answer we got from the second method with a
slightly different constant. In other
words,
We can do a similar manipulation to get the answer from the
third method. Again, starting with the
answer from the first method use the double angle formula and then substitute
in for the cosine instead of the sine using,
Doing this gives,
which is the answer from the third method with a different
constant and again we can relate the two constants by,
So, what have we learned here? Hopefully we’ve seen that constants of
integration are important and we can’t forget about them. We often don’t work with them in a Calculus I
course, yet without a good understanding of them we would be hard pressed to
understand how different integration methods and apparently produce different
answers.