Conditions for the applicability of u-substitution

[justthisonequestion]

What are the conditions for applicability of u-substitution, 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.

 

[Charles Link]

It's interesting. I haven't studied it in sufficient detail to give a complete answer, but one thing that stands out is that you don't have a one-to-one continuous mapping from $x$ to $u$. I think that is likely to be the source of the difficulty.

 

[PeroK]

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.

If $u = x^2$, then $x = \pm \sqrt{u} = \pm u^{1/2}$. In particular, the equation $x = u^{1/2}$ is not correct for all $x$ in this case.

You have, therefore, a different relationship between $x$ and $u$ depending on whether $x$ is positive or negative. So, you need to split the integral to do the substitution in this case.