Constants of Integration
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,

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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
.

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