Mathematics - Definite Integration Question with Solution | TestHub

We have${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{dx}{\left[x\right]+\left[\mathrm{s}\mathrm{i}\mathrm{n}x\right]+4}$

$={\int }_{-\frac{\pi }{2}}^0\frac{dx}{\left[x\right]+3}+{\int }_0^{\frac{\pi }{2}}\frac{dx}{\left[x\right]+4}$

$={\int }_{-\frac{\pi }{2}}^{-1}\frac{dx}{1}+{\int }_{-1}^0\frac{dx}{2}+{\int }_0^1\frac{dx}{4}+{\int }_1^{\frac{\pi }{2}}\frac{dx}{5}$

$={\left[x\right]}_{-\frac{\pi }{2}}^{-1}+{\left[\frac{x}{2}\right]}_{-1}^0+{\left[\frac{x}{4}\right]}_0^1+{\left[\frac{x}{5}\right]}_1^{\frac{\pi }{2}}=\left(-1+\frac{\pi }{2}\right)+\left(0+\frac{1}{2}\right)+\frac{1}{4}+\frac{\pi }{10}-\frac{1}{5}$

$=\frac{-20+10\pi +10+5+2\pi -4}{20}=\frac{12\pi -9}{20}=\frac{3}{20}\left(4\pi -3\right)$