Somas,diferencias e constantes

So by now you've probably seen some examples of antiderivatives. Let's just do a quick review.

Yep, so we write that the antiderivatives of $\cos x$ are of the form $\sin x + C.$

Ok! So far all we've done is review. Let's try something trickier.

So you just saw that the antiderivative of the sum of two functions is equal to the sum of their antiderivatives.

Ok! Formally, what we saw above can be expressed as:

If $f(x)$ is a function with antiderivatives in the form $F(x) + C,$ and $g(x)$ is a function with antiderivatives in the form $G(x) + C,$ then $f(x) + g(x)$ is a function with antiderivatives of the form $F(x) + G(x) + C.$

$C$ is just an arbitrary constant. There's quite a few things wrong with the above statement. Try to figure out what's wrong.

Great! If you are still confused, here is a pretty clean way to see this:

If $f(x)$ has an antiderivative $F(x)$ and $g(x)$ has an antiderivative $G(x),$ then $f(x) + g(x)$ has an antiderivative $F(x)+G(x).$ But as we saw before, we can add of subtract constants to an antiderivative, and it will still be an antiderivative, and we see that $F(x)+G(x) + C$ is the set of all antiderivatives of $f(x)+g(x)$.

Please try problems 9-12 on this worksheet. They only involve the power rule, but they are still linear. Good luck!