I'm trying to evaluate the integral
$$ \int_{-1}^1 \sqrt{(x^2+y^2)^k+B} \, \mathrm{d}y $$
WolframAlpha doesn't return a response even for simplified versions of this, but I believe it can be evaluated through substitution using polar coordinates. In particular, let the coordinate transformation be define so that
$$ x = r\cos\phi $$ $$ y = r\sin\phi $$
and
$$ r = \sqrt{x^2+y^2} $$ $$ \phi = \arctan\left(\frac{y}{x}\right) $$
From these expressions, I compute the $dy$ differential to be
$$ dy = \frac{dr}{\sin\phi} + \frac{r\,d\phi}{\cos\phi} $$
so the integral may be written
$$ \int_{y=-1}^{y=1} \sqrt{r^{2k}+B}\left(\frac{dr}{\sin{\phi}} + \frac{rd\phi}{\cos{\phi}}\right) $$
The indefinite integral of this integrand can be computed with WolframAlpha, though the result is a bit messy. I can then find the definite integral by evaluating the result at $y=1$ and $y=-1$ and taking the difference of the results. This could perhaps by simplified by noting that the original integrand is symmetric with respect to a change in the sign of $y$, so the limits of integration could be changed to $y=0$ and $y=1$ and the integral multiplied by $2$.
Are there any apparent errors in this approach to evaluating this integral?