Okay, I generally have no problems, except when the derivative of the term that needs to be substituted is a constant, say, 1 and the coefficient of the $\displaystyle dx$ or whatever in the original question is a pronumeral, say, $\displaystyle 3x$.

Here is an example:

Find the integral between $\displaystyle x=-3$ and $\displaystyle x=-2$ where $\displaystyle y=x(3+x)^7$ using the substitution $\displaystyle u=3+x$

I have found that, by differentiation, $\displaystyle du=dx=1$

My problem is:

Once i get to this stage:

$\displaystyle \int_{-3}^{-2}$ $\displaystyle (3+x)^7$ $\displaystyle x dx$

$\displaystyle x=-2$ then $\displaystyle u=1$ and

$\displaystyle x=-3$ then $\displaystyle u=0$

and the $\displaystyle (3+x)^7$ then becomes $\displaystyle u^7$

but what about the $\displaystyle x dx$?

how do I get that into $\displaystyle du$?

if the differentiated $\displaystyle u$ had an $\displaystyle x$ in it, then fine, I could do that, but since it doesn't, I'm in a pickle.

And you can only have constant terms in front of the whole integral such as $\displaystyle 1/2 $$\displaystyle \int_{-3}^{-2}......$

But what do I do for these types of integrals?