Question

$\int\limits_{0}^{5} \cos \left(\pi\left(x-\left[\frac{x}{2}\right]\right)\right) d x$ 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)

$I=\int\limits_{0}^{5} \cos \left(\pi x-\pi\left[\frac{x}{2}\right]\right) d x$
$\Rightarrow I=\int\limits_{0}^{2} \cos (\pi x) d x+\int\limits_{2}^{4} \cos (\pi x-\pi) d x+\int\limits_{4}^{5} \cos (\pi x-2 \pi) d x$
$\Rightarrow I=\left[\frac{\sin \pi x}{\pi}\right]_{0}^{2}+\left[\frac{\sin (\pi x-\pi)}{\pi}\right]_{2}^{4}+\left[\frac{\sin (\pi x-2 \pi)}{\pi}\right]_{4}^{5}$
$\Rightarrow I=0$