did u draw how it looks

?

and when u do that ... u'll see how u should do integration.... limit on x (dx) and y (dy) depending how u gonna do it... first by x or y ?

P.S. U don't need that subtitution to solve this one ...

if u change order of integration it's much much easier to solve ....

look at it like this

$\displaystyle I= \int_{0} ^{4} dy \int_{0}^{\sqrt{y}} \frac{x}{\sqrt{1+y^2}}dx$

$\displaystyle I= \int_{0} ^{4} \frac{dy}{\sqrt{1+y^2}} \int_{0}^{\sqrt{y}}xdx$

$\displaystyle I=\frac {1}{2} \int_{0} ^{4} \frac {ydy}{\sqrt{1+y^2}}$

$\displaystyle 1+y^2=t^2 \to 2ydy=2tdt \to ydy =tdt $

$\displaystyle I=\frac {1}{2} \int_{0} ^{4} \frac {dt}{t}$ ....