Let $D = \{(x,y) \in \mathbb{R}^2 \mid 0 < a \leq x \} $. Show that $$ \iint_{D} e^{-(x^2 + a^2 )} \, dx \, dy = ae^{-a^2}\int_0^\infty \frac{e^{-u^2}}{u^2 + a^2} \,du $$ using the change of variables $x^2 + y^2 = u^2 + a^2 $ and $y = vx$.
I found the Jacobian to be $\frac{u}{1+v^2}$ and the limits of integration $0 < u < \infty$ and $\frac{-u}{a} < v < \frac{u}{a}$. This leads me to $$ \iint_D e^{(x^2 + a^2 )} \, dx \, dy = \int_0^\infty \int_{\frac{-u}{a}}^{\frac{u}{a}} e^{-(u^2+a^2)}\frac{u}{1+v^2} \, dv \, du $$ I can see how $e^{-a^2}$ factors out, but if I just integrate with respect to $v$ I end up with some $\arctan\left(\frac{u}{a}\right).$
I think I have to make another another substituition and call it $u$ as well, but I couldn't think of a good one.