There are three steps to solving a math problem.
Find given that f is odd and
We need to use various properties of integrals to rearrange this equation until there's a in it somewhere. Then we solve for
There are so many ways to start this, deciding which to do is a project in itself. We'll do one way to get the answer here, and we'll do it a different way to check our work.
First, split up the integral to get
Work with the first term first. Since the first term is the integral of the constant 4 on an interval of length 12, that term is equal to 48.
Now we have
Switch the limits and the sign:
Pull out the constant:
Since f is odd, , so
Now we have
We know this is supposed to be equal to 3:
If we do the problem again, but a different way, we should get the same answer.
switch the limits and the sign first this time:
Now we'll split up the integral, but we need to be careful that the negative sign still affects everything:
Since f is odd, 3f (x) is also odd. This means , since
is zero. Put this back into our expression:
Now we pull out the constant:
We can substitute 48 for like we did before:
It's time to start solving. We know this is all supposed to equal 3.
Adding -3 and to both sides,
and again we get
which is reassuring.