Correct Answer - Option 4 : 2
\(\mathop \smallint \limits_0^{\frac{\pi }{2}} \mathop \smallint \limits_0^{\frac{\pi }{2}} \sin \left( {x + y} \right)dxdy\)
\( = \mathop \smallint \limits_0^{\frac{\pi }{2}} \mathop \smallint \limits_0^{\frac{\pi }{2}} \sin x\cos ydxdy + \mathop \smallint \limits_0^{\frac{\pi }{2}} \mathop \smallint \limits_0^{\frac{\pi }{2}} \sin y\cos xdxdy\)
\( = \mathop \smallint \limits_0^{\frac{\pi }{2}} \left[ { - \cos x\cos y} \right]_0^{\frac{\pi }{2}}dy + \mathop \smallint \limits_0^{\frac{\pi }{2}} \left[ {\sin y\sin x} \right]_0^{\frac{\pi }{2}}dy\)
\( = \mathop \smallint \limits_0^{\frac{\pi }{2}} \cos ydy + \mathop \smallint \limits_0^{\frac{\pi }{2}} \sin ydy\)
\( = \left[ {\sin y} \right]_0^{\frac{\pi }{2}} + \left[ { - \cos y} \right]_0^{\frac{\pi }{2}} = 2\)
