Solving an integral when using a dummy variable I'm trying to solve an integral and I don't remember how to when a variable is in the integral symbol, if that makes sense. I'm not sure what the correct terminology is.
So say I have an integral like:
$$f(x) = \int_{-3}^{x^{3}} t^2 dt$$
How do I solve it? What do I have to do differently?
 A: When you are computing the definite integral
$$\int_a^b f(t)dt$$
your usual procedure has been:
(i) Find an antiderivative $F(t)$ of $f(t)$.  That means, find a function $F(t)$ such that $F'(t)=f(t)$.
(ii) Then your "answer" is $F(b)-F(a)$.
The procedure is exactly the same if $a$ and $b$ are not constants, but are functions of some variable $x$.  So (ii) remains unchanged.  
In your particular integral, we have $a=a(x)=x^3$ and $b=b(x)=-3$.  The "lower" limit of integration is a constant, which you can think of as a constant function, and the upper limit is the function $x^3$.
Let's calculate.  We have $f(t)=t^2$.  So one antiderivative of $f(t)$ is $F(t)$,
where 
$$F(t)=\frac{t^3}{3}$$
Now do the familiar substitution process.  We get
$$\int_{-3}^{x^3}t^2dt=F(x^3)-F(-3)=\frac{(x^3)^3 -(-3)^3}{3}$$
We might want to simplify this to
$$\frac{x^9+27}{3}$$
Comments: The variable $t$ is called a "dummy variable" roughly because ultimately it plays no role in the answer.  You would get exactly the same thing if you had $u$ as the variable, or $w$, instead of $t$.
In principle, you could also have used the letter $x$ as the dummy variable.  But don't ever do it when one or both of the limits of integration involves $x$.  Although it is technically not wrong, the chances of getting confused are too high.
A: To be explicit, you can do this
$$f(x) = \int_{-3}^{x^3} t^2\, dt = {t^3\over 3}\bigg|_{-3}^{x^3} = {x^9 + 27\over 3}.$$
There is nothing scary here; just treat $x$ as if it were a constant and you will be fine.
A: Easily done; let $u=x^3$, proceed with the integration as usual, and then replace the $u$ in your result with $x^3$.
