### Video Transcript

Determine the definite integral
from negative one to one of nine 𝑥 cubed minus six 𝑥 all over two 𝑥 squared plus
nine with respect to 𝑥.

In this question, we are asked to
evaluate the definite integral of a rational function. We could do this by attempting to
find an antiderivative. However, it is often possible to
simplify definite integrals by using the properties of integration. We can evaluate this definite
integral by recalling that the definite integral from negative 𝑎 to 𝑎 of an odd
function 𝑓 of 𝑥 with respect to 𝑥 is zero for any constant 𝑎.

The reason that this result holds
true is that the graphs of odd functions have rotational symmetry of 180 degrees
about the origin. So the area under the curve to the
left and right of zero for an equal distance will be the same.

This is a useful result in this
case because the given integrand is an odd function. We can show this in many ways. For instance, we can recall that we
say a function is odd if 𝑓 of negative 𝑥 equals negative 𝑓 of 𝑥 for any value of
𝑥 in the domain of 𝑓. We can then note that the numerator
of the integrand is a polynomial containing only odd powers. We can recall that this means that
the numerator is an odd function. Similarly, we can see that the
denominator is a polynomial only containing even powers of 𝑥. We can recall that this means that
it is an even function.

Finally, we know that the quotient
of an odd and even function is odd, so the integrand is odd. We can also show that the integrand
is odd more directly by substituting negative 𝑥 into the function to obtain the
following expression. We then see that the even powers
will lose the power of negative one and the odd powers will retain this factor.

We can then take out the factor of
negative one in the numerator and simplify the denominator to obtain the following
expression. We see that this is equivalent to
negative 𝑓 of 𝑥. Hence, the integrand is odd. Since we have shown that the
integrand is odd, we can apply this property to conclude that the definite integral
from negative one to one of nine 𝑥 cubed minus six 𝑥 all divided by two 𝑥 squared
plus nine is zero.