double integral of (x^2*tanx+y^3+4)dR. Over the region x^2+y^2(greater than or equal to)2

our teacher instructed us not to change the variables into polar coordinates, so I sticked to using the Cartesian coordinate system. Limits of integration since circle is symmetric to both axis it could be interchanged, I made my limits as

-(2)^(1/2)<x<(2)^(1/2),

-(2-x^2)^(1/2)<y<(2-x^2)^(1/2)..

So as I said I separately integrated the variables, with "y^3" going to zero and "4" turning into 8pi.

But my problem is about the double integral of x^2*tanxdR. At first , I integrated it with respect to y,

so the result is Integral of (2)((2-x^2)^(1/2)*x^2*tanx) dx.

But at this point, I don't know what to do. I couldn't figure out how to take the anti derivative , even Wolfram could not give me the anti derivative. But it could give me the answer to this definite integral that is 0. Not allowed to use taylor series either,

Thanks in advance for your help.