<s>[INST] Calculte indefinte integral of $\int \frac{dx}{\sqrt{(x+2)(3-x)}}$ I have to calculate $$\int \frac{dx}{\sqrt{(x+2)(3-x)}}$$.
I tried to use -
$\int u'v = uv - \int v'u$, but im pretty stuck.
Thanks.
 [/INST] \begin{align}
\int \frac{dx}{\sqrt{(x+2)(3-x)}}
=& \ \text{sgn} (\frac12-x)\int \frac{d\left(\sqrt{(x+2)(3-x)}\right)}{\sqrt{\frac{25}4- (x+2)(3-x)}}\\
=& \ \text{sgn} (\frac12-x) \sin^{-1}\bigg(\frac25 \sqrt{(x+2)(3-x)}\bigg)+C
\end{align}
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