Correct use of the Substitution Rule (or Integration by Substitution) When using the Substitution Rule (or Integration by Substitution), Stewart (Calculus: Early Transcendentals, 8e, 2016, p. 413) writes $$\int\sqrt{1+x^2}2x\,dx\overset{1}{=}\int\sqrt{u}\,du.$$
But strictly speaking, is $\overset{1}{=}$ correct?
I thought $u$ was just a dummy variable that can be replaced by any other letter, e.g. $x$. In which case, $$\int\sqrt{u}\,du=\int\sqrt{x}\,dx=\frac{2}{3}x^{3/2}+C,$$ which is clearly not equal to $\int\sqrt{1+x^2}2x\,dx$.
And if $\overset{1}{=}$ is wrong, then what is the correct way to use the Substitution Rule?

$x$ is not a dummy variable, as that is an indefinite integral. You are likely thinking of definite integrals, in which case rewriting the integral using the same variable is fairly common, as it would indeed be a dummy variable:
$$\int_0^12x\sqrt{x^2+1}~\mathrm dx=\int_1^2\sqrt x~\mathrm dx$$
since the variables vanish after performing the integration. The same is not true for indefinite integrals, where the variable remains after integration:
$$\int2x\sqrt{x^2+1}~\mathrm dx=\frac23(x^2+1)^{3/2}+C_1=\frac23u^{3/2}+C_1=\int\sqrt u~\mathrm du\ne\int\sqrt x~\mathrm dx=\frac23x^{3/2}+C_2$$
where $u=x^2+1$.
