Question

$\underset{0}{\overset{5}{\int }}\cos \left(\pi \left(x-\left[\frac{x}{2}\right]\right)\right)dx$ Where $[t]$ denotes greatest integer less than or equal to $t$, is equal to:

• A. -3
• B. -2
• C. 2
• D. 0

Answer 1

Answer: (D)

$I=\underset{0}{\overset{5}{\int }}\cos \left(\pi x-\pi \left[\frac{x}{2}\right]\right)dx$
$\Rightarrow I=\underset{0}{\overset{2}{\int }}\cos (\pi x)dx+\underset{2}{\overset{4}{\int }}\cos (\pi x-\pi )dx+\underset{4}{\overset{5}{\int }}\cos (\pi x-2\pi )dx$
$\Rightarrow I={\left[\frac{\sin \pi x}{\pi }\right]}_0^2+{\left[\frac{\sin (\pi x-\pi )}{\pi }\right]}_2^4+{\left[\frac{\sin (\pi x-2\pi )}{\pi }\right]}_4^5$
$\Rightarrow I=0$