<s>[INST] Integration of root of quadratic over quadratic I need help to get a generalized solution for this type equations:
$$ \Gamma(v) = \int\frac{(1-v^2)^{\frac{1}{2}}}{a^{2}-b^{2}v^{2} }dv $$
NOTE: This is not a homework question. I am in high school and I was reading up some extra material and landed on this from another equation.
Sorry for the bad formatting.
Thanks
 [/INST] Substituting $v=\sin\theta$ gives $\sqrt{1-v^2}=\cos\theta$, $dv=\cos\theta\,d\theta$ so we have
\begin{eqnarray}
\int\dfrac{\cos^2\theta}{a^2-b^2\sin^2\theta}\,d\theta&=\int\dfrac{1-\sin^2\theta}{a^2-b^2\sin^2\theta}\,d\theta\\
&=\dfrac{1}{b^2}\int\left[1-\dfrac{a^2-b^2}{2a}\left(\frac{1}{a-b\sin\theta}+\frac{1}{a+b\sin\theta} \right)  \right]\,d\theta
\end{eqnarray}
The rational functions of $\sin\theta$ can be completed by the substitution
\begin{equation}
t=\tan\left(\frac{\theta}{2} \right),\, d\theta=\frac{2dt}{1+t^2},\,\sin\theta=\frac{2t}{1+t^2}
\end{equation}
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