$$ u = 2x+1 $$
$$ du = 2 dx $$
$$ dx = \frac{1}{2} du $$
Upper Bound
$$x=2$$
$$u = 2(2)+1$$
$$u = 5$$
Lower Bound
$$ x=0 $$
$$ u = 2(0)+1 $$
$$ u = 1 $$
Substituting these values into the definite integral:
$$ \int\limits_0^2 \sqrt{\colorbox{aqua}{$2x+1$}} \colorbox{yellow}{$dx$} $$
$$ \int\limits_1^5 \sqrt{\colorbox{aqua}{$u$}} \cdot \colorbox{yellow}{$\frac{1}{2} du$} $$
$$ = \boxed{\frac{1}{2}\int\limits_1^5 \sqrt{u} du} $$