Basic integral question & sin, tan, cos.

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

Re: Basic integral question & sin, tan, cos.

$\displaystyle \frac{x^3}{3}=\frac{1\cdot x^3}{3\cdot1}=\frac{1}{3}\cdot\frac{x^3}{1}=\frac{ 1}{3}x^3$

If you look at the graph of $\displaystyle y=-\cos(x)+C$ (choose any $\displaystyle C$) and then simultaneously plot the graph of $\displaystyle y=\sin(x)$ you will see that the second graph represents the instantaneous rate of change of the first.

Re: Basic integral question & sin, tan, cos.

But that's just it, what is -cos, cos, tan, sin etc? Are they actual constants like PI=3.14 etc? Would you be able to plot a graph $\displaystyle y=\-cos(x)+4$ just so I can see what value -cos represents.

Re: Basic integral question & sin, tan, cos.

I've just been looking at a cheat chart and I think something like $\displaystyle y = -cos(60)+4$ the $\displaystyle -cos(60)$ would represent $\displaystyle -\frac{1}{2}$ ?

Re: Basic integral question & sin, tan, cos.

I'm sorry, you stated you have a basic understanding of the trigonometric functions, so I naturally assumed that you did.

There are many sites online that will graph functions, and to essentially teach trigonometry is beyond the scope for me in online posts. There is a great deal of online material though that can get you started. Just do a search on trigonometry.

I highly recommend you learn this before exploring the calculus!

Re: Basic integral question & sin, tan, cos.

Oh ok no problem, I thought this was still algebra, I must have been getting a bit carried away :D

Re: Basic integral question & sin, tan, cos.

I would recommend checking out Khan Academy there are videos that you can watch for trigonometry and also practice problems for this topic.