# 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

Staff Emeritus
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

Staff Emeritus
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.