Find: $\displaystyle \int \frac{x^{2}}{\left(x^{2}+1\right)\left(3 x^{2}+4\right)} \mathrm{d} x$ OR Evaluate: $\displaystyle \int_{-2}^{1} \sqrt{5-4 x-x^{2}} \mathrm{~d} x$

\[\int \frac{x^{2}}{(x^{2}+1)(3x^{2}+4)}\,dx=\int -\frac{1}{x^{2}+1}\,dx+\int \frac{4}{3x^{2}+4}\,dx \]\[= -\tan^{-1}x+\frac{4}{3}\times\frac{\sqrt{3}}{2}\tan^{-1}\frac{\sqrt{3}x}{2}+C \]\[= -\tan^{-1}x+\frac{2}{\sqrt{3}}\tan^{-1}\frac{\sqrt{3}x}{2}+C \]OR(b)\[\int_{-2}^{1}\sqrt{5-4x-x^{2}}\,dx \]\[=\int_{-2}^{1}\sqrt{9-(x+2)^{2}}\,dx \]\[=\left[\frac{x+2}{2}\sqrt{9-(x+2)^{2}}+\frac{9}{2}\sin^{-1}\frac{x+2}{3}\right]_{-2}^{1} \]\[= 0+\frac{9}{2}\cdot\frac{\pi}{2}-0 \]\[=\frac{9\pi}{4} \]