If $\varphi(t)=\left\{\begin{array}{l}1, \text { for } 0 \leq t+1 \\ 0 \text { othenwe then }\end{array} \int_{-3000}^{3000}\left(\sum_{r=2014}^{2016} \varphi\left(t-r^{\prime}\right) \varphi(t-2016)\right) d t=\right.$

Question

If $\varphi(t)=\left\{\begin{array}{l}1, \text { for } 0 \leq t<1 \\ 0 \text { othenise then }\end{array} \int_{-3000}^{3000}\left(\sum_{r=2014}^{2016} \varphi\left(t-r^{\prime}\right) \varphi(t-2016)\right) d t=\right.$
(A) a real number
(B) 1
(C) 0
(D) does not exist

Answer 1

Ans : $(A, B)$
 $\int_{-3000}^{3000} \varphi(t-2016)(\varphi(t-2014)+\varphi(t-2015)+\varphi(t-2016)) \cdot d t$
$$\int_{-3000}^{2016} 0 . d t+\int_{2016}^{2017} 1 \cdot(0+0+1) \cdot d t+\int_{2017}^{3000} 0 \cdot d t=1$$

Answer 2

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Answer 3

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