moving constants outside integrals
Explanation
It's frequently useful to move constants outside of an integral before
looking for an antiderivative. As an example, consider the
integral
ò
7 x5 dx.
We can rewrite this as
7 ò
x5 dx
which lets us ignore the constant multiple "7" while we do the
integral:
ò
x5 dx =
(1/6) x6 + c
(by
basic antidifferentiation). This makes
the original integral easy:
ò
7 x5 dx =
7 [ (1/6) x6 + c ].
Or, stepping all the way through the problem,
ò 7
x5 dx |
= |
7 ò
x5 dx |
|
= |
7 [ (1/6) x6 + c ] |
|
= |
(7/6) x6 + C |
(there's the smallest of subtleties going on with the constants
C and c here: c is an arbitrary constant, as is
C, with C = 7 c).
Why would we want to do this?
By not considering constants while we find an antiderivative we reduce
the number of things that we need to keep track of in the integral.
This will often reduce the complexity of the calculation that
we have to do---in the example above, we were able to only think about
the factor of (1/6) required in the antidifferentiation rather than
having to also remember to keep track of the factor of 7 (until the
very end, when it's easy).
Would we ever not want to do this?
We might not want to do this when the constant that we're moving
outside of the integral is one which we can use in finding the
antiderivative. The simplest example of this is something like
ò
3 x2 dx =
x3 + C,
for which we would end up introducing a factor of (1/3) if we didn't
consider the constant multiple of three.
Can we ever not do this?
No.
int tutorial: constants
Last Modified: Fri Aug 31 11:23:11 EDT 2001
Comments to
glarose@umich.edu
©2001 Gavin LaRose, UM Math Dept.