How to perform U substitution when there is no direct dx to du mapping? I have been asked to solve this problem using a change of variable and a power series expansion of the exponential term:
$\frac{1}{\sqrt(2\pi)} \int_0^\sqrt2 e^{-\frac{x^2}{2}} dx$
I expanded the exponential term to the Taylor Series:
$e^{-\frac{x^2}{2}} \approx 1 - \frac{x^2}{2} + \frac{x^4}{8} - \frac{x^6}{48} + \frac{x^8}{384}$
This then leaves me with this:
$\frac{1}{\sqrt(2\pi)} \int_0^\sqrt2 1 - \frac{x^2}{2} + \frac{x^4}{8} - \frac{x^6}{48} + \frac{x^8}{384} dx$
I interpreted the "change of variable" as a reference to U substitution, but I am stuck when trying to substitute something in either the original formula or in the expansion of it.
I tried substituting $u = x^2$ in the revised formula:
$u = x^2$
$du = 2x$
$\frac{1}{\sqrt(2\pi)} \int_0^2 1 - \frac{u}{2} + \frac{u^2}{8} - \frac{u^3}{48} + \frac{u^4}{384} ??dx??$
But I am stuck on how to convert dx to du
I calculated that $du = 2x dx$, and $x = \sqrt u$, but how do I replace that into the formula? Is that even the right question to ask?

You decided to use
$$x^2=u \implies x=\sqrt u\implies dx=\frac{du}{2 \sqrt{u}}$$
