If a is positive integer then the number of values of 'a' satisfying 

$\underset{0}{\overset{\pi /2}{\int }}\left\{a^2\left(\frac{\cos 3x}{4}+\frac{3}{4}\cos x\right)+a\sin x-20\cos x\right\}dx\le \frac{a^2}{3}$ are 

$$\begin{gathered} \underset{0}{\overset{\mathrm{\pi }/2}{\int }}\left[{\mathrm{a}}^2\left(\frac{\cos 3\mathrm{x}}{4}+\frac{3}{4}\cos \mathrm{x}\right)+\mathrm{a} \sin \mathrm{x}-20\cos \mathrm{x}\right] \mathrm{dx}\le \frac{-{\mathrm{a}}^2}{3} \\ \Rightarrow {\mathrm{a}}^2{\left[\frac{\sin 3\mathrm{x}}{12}\right]}_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\frac{3{\mathrm{a}}^2}{4}{(\sin \mathrm{x})}_0^{\mathrm{\pi }/2}+\mathrm{a}{(-\cos \mathrm{x})}_0^{\mathrm{\pi }/2}-20 \sin \mathrm{x}{]}_0^{\mathrm{\pi }/2}\le \frac{-{\mathrm{a}}^2}{3} \\ \Rightarrow {\mathrm{a}}^2 \left[\frac{-1}{12}+\frac{3}{4}\right]+\mathrm{a}\left[1\right]-20\le \frac{-{\mathrm{a}}^2}{3} \\ \Rightarrow {\mathrm{a}}^2\left(\frac{2}{3}\right)+\mathrm{a}-20\le \frac{-{\mathrm{a}}^2}{3} \\ \Rightarrow {\mathrm{a}}^2+\mathrm{a}-20\le 0 \\ \Rightarrow -5\le \mathrm{a}\le 4 \\ \mathrm{a}=1, 2, 3, 4 \left(\therefore \mathrm{a}>0\right) \end{gathered}$$