Question

Let \( [t] \) denote the greatest integer less than or equal to \( t \). Then the value of the  integral \(\int\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x\) is equal to

Answer 1

\(I = \int\limits_{-3}^{101} ([\sin(\pi x)] + e^{[\cos (2 \pi x)]}) dx\)

[sin πx] is periodic with period 2 and e^[cos(2πx)] is periodic with period 1.

So,

\(I = 52\int\limits_{0}^{2} ([\sin(\pi x)] + e^{[\cos (2 \pi x)]}) dx\)

\(= 52 \left\{ \int \limits_1^2 - 1dx + \int \limits _{\frac 14}^\frac 34 e^{-1}dx + \int \limits_{\frac 54}^ \frac 74e^{-1}dx+ \int \limits_{\frac 14}^0e^{0}dx+ \int \limits_{\frac 34}^\frac54e^{0}dx+ \int \limits_{\frac 74}^2e^{0}dx\right\}\)

\(= \frac {52}e\)