Refer to the description below. Are there any other anti-derivatives of f(x) = 12x^2 + 2x, and g(x) = cos x. (Cite another anti-derivative for each function.)

Question

There are already two known antiderivatives of each function, specifically:

1.) f(x) = 12x^2 + 2x:

Antiderivatives:

4x³+x²

4x³+x²-1

2.) g(x)=cos x

Antiderivatives:

sin x

sin x+π

Now, what are the other antiderivatives of each function? (Just one new antiderivative for each function and also state how you got it)

Answer 1

Given one antiderivative, you can always find infinitely more by adding arbitrary constants - recall that the derivative of a constant is 0, and so adding a constant does not change the derivative at all.

So, for example, 4x^3+x^2-2 is another (I chose -2 - again, literally any constant you pick will also work)