Mathematics - Definite Integration Question with Solution | TestHub

$I={\int }_0^5\cos \left(\left[\frac{x}{2}\right]\pi -\pi x\right)dx$

$\because x\in \left(0,5\right) \therefore x\in \left(0,\frac{5}{2}\right)\Rightarrow \left[\frac{x}{2}\right]\in \left\{0,1,2\right\}$

$I={\int }_0^2\cos \left(-\pi x\right)dx+{\int }_2^4\cos \left(\mathrm{\pi }-\pi x\right)dx+{\int }_4^5\cos \left(2\mathrm{\pi }-\pi x\right)dx$

$\Rightarrow I={\left[\frac{\mathrm{sin\pi }x}{\mathrm{\pi }}\right]}_0^2+{\left[-\frac{\mathrm{sin\pi }x}{\mathrm{\pi }}\right]}_2^4+{\left[\frac{\mathrm{sin\pi }x}{\mathrm{\pi }}\right]}_4^5=0$