I've just been looking at some algebra called intergrals.

$\displaystyle f(x) = 3x^4+x^2-9x+1 = F(x)\frac{3x^5}{5}+\frac{x^3}{3}-\frac{9x^2}{2}+x+c$

I understand this fully but in the next step when he simplifies it a little he turns the $\displaystyle \frac{x^3}{3}$ into $\displaystyle \frac{1}{3}x^3$

I don't understand why he put a 1 there in place of the $\displaystyle x^3$. Where did the 1 come from? Also one other question I have which I have is:

$\displaystyle f(x)= \sin x = F(x)= -\cos x+c$

I have a basic understand of sin, cos and tan but what exactly do the represent when used in equations?

Can someone just explain what these equation actually means? Thanks