Integral problem w/ both limits as variable

Hey all, I'm very new to these forums. I'm not looking for the answers, but rather to figure out if the way I go about doing this problem is correct or if I am totally wrong any explanation would be very appreciated.

Question: Use the fundamental theorem of Calculus to find the derivative of the given function

$\displaystyle \int_{x^3}^{x} cos(t^3)dt$

What I did:

$\displaystyle \int_{x^3}^{x} cos(t^3)dt$ = $\displaystyle \int_{0}^{x} cos(t^3)dt$ - $\displaystyle \int_{0}^{x^3} cost(t^3)dt$ = $\displaystyle cos(x^3) - cos(x^9) * 3x^2$

**Ive never used LaTeX before so I hope this is the right way of showing it.

**If I messed up the whole thing or any part please give me tips on where I went wrong, thank you!

P.S - is there any online program that I could input an integral such as this so I can check myself?

Sorry for all the questions + thank you!