Integration by substitution gone wrong

You should take a look at Michael Spivak's Calculus, where the substitution theorem has exactly this requirement (injective substitution), and a discussion of the problem you ask about. (That discussion may take place in one or more of the many excellent problems in the integration chapter; I can't remember and don't have my copy with me.)

Well they are adressed. See for an example http://faculty.swosu.edu/michael.dougherty/book/chapter07.pdf.

The easiest way to avoid this problem is to choose a bijective-substitution. Eg one that is one-to-one and has an inverse.

In most cases a substitution rule is introduced for integrals of the specific form $$\int _{{a}}^{{b}}f(\varphi (t))\cdot \varphi '(t)\,{\mathrm {d}}t$$

where $f:I \to \Bbb R$ is continuous and $\varphi: [a,b] \to I$ is continuously differentiable. Then it holds $$\int _{{a}}^{{b}}f(\varphi (t))\cdot \varphi '(t)\,{\mathrm {d}}t=\int _{{\varphi (a)}}^{{\varphi (b)}}f(x)\,{\mathrm {d}}x$$

If your integrand is $t^n$ for $n$ even you wont find $f$ and $\varphi$ s.t.

$$f(\varphi (t))\cdot \varphi '(t) = t^n$$

But if $n$ is odd you can choose $\varphi(t) = t^2$ and $f(t) = \frac{1}{2}t^\frac{n-1}{2}$ and the substitution works… so there is no need to consider integrals of your form because you cannot use subsitution rule for them because they don't satisfy the given assumptions.