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*i.e.*when does it not work? Note that I'm not asking when is it a bad idea (that won't get you any closer to evaluating the integral), but are there any conditions that cause u-sub to yield wrong answers?

I started running into what I think is a case of u-sub not working when I was thinking about integrals of odd functions, ex:

$$\int_{-\infty}^{\infty}x e^{-a x^2}dx$$

$$u=x^2$$

$$du=2xdx$$

$$\int_{-b}^{b}x e^{-a x^2}dx=\frac{1}{2}\int_{b^2}^{b^2} e^{-a u}dx=0$$

Where the last part equals zero because now the bounds are equal...

But this begged the question... why can't I just u-sub in such a way that the bounds on the integral are always equal, and all integrals go to 0? - Obviously there has to be some constraint on the applicability of u-sub.

Here is an example where it seems u-sub just leads to the wrong answer...

$$\int_{-2}^{1}x^4dx=\frac{33}{5}=6.6$$

With $$u=x^2$$ we get:

$$\frac{1}{2}\int_{4}^{1}u^{3/2}du=-\frac{31}{5}=-6.2$$

So what gives? What basic mathematical principle is being violated here? I'm sure I learned this at some point... waaay back when.