polynomial integral concept question: a term removed from polynomial, but integral is the same.

Question

Sorry I have no better name for this question.

while doing homework I came across this equation:

And I am completely baffled on how this works. So can someone explain to me why they are equal because I simply do not see how they can be related in any ways.

Also, will this property remain if the degree of the removed term is greater or less than 1, or if it is another type of function.

Thanks.

Answer 1

$ax^4+c$ is symmetric either side of $x=0$. It is called an even function. So the area under the curve is the same on either side, and the total area equals twice the area on the right-hand side.
$bx$ is an odd function, because the value at $-x$ is the negative of the value at $+x$. So the integral for negative $x$ cancels out the integral for positive $x$. That is why it disappears from the formula.