Answer
See explanation below.
Work Step by Step
The main reason for applying the rule of substitution is to obtain an integral of less complexity, with a new variable which would be easier to solve than the initial integral.
After solving with the new variable of integration, we back-substitute in order to convert the function set to be defined with the original variable.
A substitution $u=u^{2}+1$ is wrong for several reasons, but the main reason is that it does not introduce a new variable.
A proper substitution would be
$ t=u^{2}+1,\quad$or$\quad v=u^{2}+1,$
or any other that introduces a new variable.
