Hold on.

- For sqrt(x^{2}+1) it is very useful to use a u-substitution, but not necessary as long as you can keep track of Chain Rule.

- Agreed. There is no purpose for outside brackets.

- The antiderivative of x^-1/2 isnotln|sqrt(x)|. It is 2 * x^(1/2) using Exponent Rule. (common misconception of antiderivatives involving ln)