Change of variables in an integral

1. Jul 10, 2015

2sin54

This is not really a homework or anything, just found myself hitting a wall when doodling around.
If I have an integral like

$\int_{-1}^{0} x(x^2-1) dx$

and I introduce a new variable:
$u = x^2$
How do I calculate the limits of the new integral? In this case the upper limit is obviously 0, but what about the lower limit?

2. Jul 10, 2015

Michael David

For the lower limit you plug in -1 in that u=x^2 and so you have u=1... same thing for the upper limit. Of course you don't necessarily have to change the variable for this integral......

3. Jul 10, 2015

HallsofIvy

When x= -1, u= (-1)2= 1, of course. If it bothers you that the lower limit of integration is larger than the upper limit just use the fact that $\int_a^b f(x)dx= -\int_b^a f(x)dx$.

4. Jul 10, 2015

2sin54

Argh, that was a brain fart.. But what about a case where
$\int_{-1}^{0} x dx$
and I make a change of variables like:
$u^2 = x$
Does the lower limit become a complex number?

5. Jul 11, 2015

HallsofIvy

Yes, and, in fact, you would have to consider the case of u= i and u= -i separately. Further you would have to consider that in the complex numbers there are an infinite number of paths between i and 0 or between -i and 0! Not every substitution is a good idea.