<s>[INST] Compute definite integral Question: Compute 
$$\int_0^1 \frac{\sqrt{x-x^2}}{x+2}dx.$$
Attempt: I've tried various substitutions with no success. Looked for a possible contour integration by converting this into a rational function of $\sin\theta$ and $\cos \theta$.
 [/INST] Let $x=\frac{1}{2}\sin t+\frac{1}{2}$, then we have
\begin{align*}
\int_0^1\frac{\sqrt{x-x^2}}{x+2}dx&=\int_0^1\frac{\sqrt{\left(\frac{1}{2}\right)^2-\left(x-\frac{1}{2}\right)^2}}{x+2}dx\\
&=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\sqrt{\left(\frac{1}{2}\right)^2-\left(\frac{1}{2}\sin t\right)^2}}{\frac{1}{2}\sin t+\frac{1}{2}+2}\frac{1}{2}\cos t\,dt\\
&=\frac{1}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos t}{\sin t+5}\cos t\,dt\\
\end{align*}
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