[tex]\displaystyle\int_{x=0}^{x=1}\int_{y=1}^{y=x}\cos y^2\,\mathrm dy\,\mathrm dx[/tex]
Change the order of integration. The region over which you're integrating can be equivalently described by the set of points in the plane, [tex]\{(x,y)\mid0\le x\le y,0\le y\le1\}[/tex].
Then the integral becomes
[tex]\displaystyle\int_{y=0}^{y=1}\int_{x=0}^{x=y}\cos y^2\,\mathrm dx\,\mathrm dy=\int_{y=0}^{y=1}y\cos y^2\,\mathrm dy[/tex]
Substitute [tex]z=y^2[/tex], [tex]\mathrm dz=2y\,\mathrm dy[/tex]:
[tex]\displaystyle\frac12\int_{z=0}^{z=1}\cos z\,\mathrm dz=\frac12\sin1[/tex]
