Question

Let $f: R \rightarrow R$ be a function defined as
$f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in R ,$
where [t] is the greatest integer less than or equal to $t$. If $\displaystyle\lim _{x \rightarrow-1} f(x)$ exists, then the value of $\int_0^4 f(x) d x$ is equal to :

• A. $-1$
• B. $-2$
• C. 1
• D. 2

Answer 1

Answer: (B)

$\displaystyle\lim _{x \rightarrow-1^{+}} a \sin \left(\pi \frac{[x]}{2}\right)+[2-x]=-a+2$
$ \displaystyle\lim _{x \rightarrow-1^{-}} a \sin \left(\pi \frac{[x]}{2}\right)+[2-x]= $
$\displaystyle\lim _{x \rightarrow-1} f(x) \text { exist when } a=-1$
Now,
$ \int\limits_0^4 f(x) d x=\int\limits_0^1 f(x) d x+\int\limits_1^2 f(x) d x+\int\limits_2^3 f(x) d x+\int\limits_3^4 f(x) d x $
$ =\int\limits_0^1(0+1) d x+\int\limits_1^2(-1+0) d x+\int\limits_2^3(0-1) d x+$
$ \int\limits_3^4(1-2) d x $
$ =1-1-1-1=-2$