Question

If $[.]$ represents the greatest integer function, then the value of $\left|\underset{0}{\overset{\sqrt{\frac{\pi }{2}}}{\int }}\left[\right[x^2]-\cos x]dx\right|$ is

Answer 1

$\underset{0}{\overset{\sqrt{\frac{\pi }{2}}}{\int }}\left[\right[x^2]-\cos x]dx=\underset{0}{\overset{1}{\int }}[-\cos x]dx+\underset{1}{\overset{\sqrt{\frac{\pi }{2}}}{\int }}[1-\cos x]dx=\underset{0}{\overset{1}{\int }}-1dx+\underset{1}{\overset{\sqrt{\frac{\pi }{2}}}{\int }}0dx=-1\therefore \left|\underset{0}{\overset{\sqrt{\frac{\pi }{2}}}{\int }}\left[\right[x^2]-\cos x]dx\right|=1$