[m]∫∫_{D}y^2cos\frac{xy}{2} dxdy= ∫_{0} ^{\sqrt{\frac{π}{2}}}(∫_{0} ^{2y}y^2cos\frac{xy}{2} dx)dy=[/m]
[blue][m]d(\frac{xy}{2}))=(\frac{xy}{2}))`_(x)dx=\frac{y}{2}dx[/m] ⇒ [m]dx=\frac{2}{y}d(\frac{xy}{2}))[/m][/blue]
[m]=∫_{0} ^{\sqrt{\frac{π}{2}}}y^2\cdot \frac{2}{y}(∫_{0} ^{2y}cos\frac{xy}{2} d(\frac{xy}{2}))dy=[/m]
[m]=∫_{0} ^{\sqrt{\frac{π}{2}}}2y(sin\frac{xy}{2})|_{0} ^{2y} dy=[/m]
[m]=∫_{0} ^{\sqrt{\frac{π}{2}}}2y\cdot (sin\frac{2y\cdot y}{2}- sin 0)dy=[/m]
[blue][m]dy^2=2ydy[/m][/blue]
[m]=∫_{0} ^{\sqrt{\frac{π}{2}}}siny^2d(y^2)=(-cosy^2)|_{0} ^{\sqrt{\frac{π}{2}}}=-cos\frac{π}{2}+cos0=-0+1=1[/m]