Question

If [.] denotes the greatest integer function and sgn denotes the signum function, then match the following columns:

Answer 1

A)
$A={\int }_{-1}^1[x+[x+\left[x\right]]]dx$
$A={\int }_{-1}^0[x+[x+\left[x\right]]]dx$ $+{\int }_0^1[x+[x+\left[x\right]]]dx$
For $-1<x<0$, $[x+[x+\left[x\right]\left]\right]=-3$
For $0\le x<1$,
$[x+[x+\left[x\right]\left]\right]=0$
$A=-3+0=-3$

B)

$B={\int }_2^5(\left[x\right]+[-x])dx$
Using $\left[x\right]+[-x]=-1$, we get
$B=-1{\int }_2^{5}dx=-3$
C)

$I={\int }_{-1}^{3}sgn(x-\left[x\right])dx$

$I={\int }_{-1}^{3}1.dx=3+1=4$

D)

$25{\int }_0^{\frac{\pi }{4}}({\tan }^6(x-\left[x\right])+{\tan }^4(x-\left[x\right]))dx$

$I=25\int 0^{\frac{\pi }{4}}({\tan }^{6}x+{\tan }^{4}x)dx=25{\int }_0^{\frac{\pi }{4}}\left({\tan }^{4}x({\tan }^{2}x+1)\right)dx$

$I=25{\int }_0^{\frac{\pi }{4}}\left({\tan }^{4}x{\sec }^{2}x\right)dx=25{\left[\frac{{\tan }^{5}x}{5}\right]}_0^{\frac{\pi }{4}}=5$