How did the u²=x part change to 2udu

that's a great question for clarifying the substitution process!

Let's break down how $u^2 = x$ changes to $2u \, du = dx$.

When we have a substitution like $u = \sqrt{x}$, we often square both sides to get rid of the square root, which gives us:

$$u^2 = x$$

Now, to find the relationship between $du$ and $dx$, we differentiate both sides of this equation.

1. Differentiate the left side ($u^2$) with respect to $u$:

   $$\frac{d}{du}(u^2) = 2u$$

   So, the differential is $2u \, du$.

2. Differentiate the right side ($x$) with respect to $x$:

   $$\frac{d}{dx}(x) = 1$$

   So, the differential is $1 \, dx$, or simply $dx$.

By equating these differentials, we get:

$$2u \, du = dx$$

This step is crucial for replacing $dx$ in the original integral with an expression involving $u$ and $du$.