Chapter 5.3, Problem 142E

In the following exercises, evaluate the double integral ${\displaystyle \underset{R}{\iint }f\left(x,y\right)dA}$ over the polar rectangular region D.

142. ${\displaystyle \underset{D}{\iint }{e}^{x^2+y^2}\left[1+2\arctan \left(\frac{y}{x}\right)\right]dA,D}=\left\{\left(r,\theta \right)|1\le \theta \le \frac{\pi }{3}\right\}$