Let [t] denote the greatest integer less than or equal to t.

Question

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\displaystyle\int_{-3}^{101}\left([\sin(\pi x)]+e^{[\cos(2\pi x)]}\right)dx$ is equal to

Accepted Answer

$[\sin\pi x]$ has period $2$ and $e^{[\cos2\pi x]}$ has period $1$; the combined integrand has period $2$, and $101-(-3)=104=52\times2$:

$$I = 52\int_0^2\left([\sin\pi x]+e^{[\cos2\pi x]}\right)dx$$

Breaking into subintervals where each bracket is constant (using $[\sin\pi x]=-1$ on $(1,2)$ except a set of measure zero where it's $0$, and $[\cos2\pi x]=-1$ near $x=1/4,3/4,\ldots$, $=0$ elsewhere):

$$I = 52\left[\int_1^2(-1)\,dx + \int_{1/4}^{3/4}e^{-1}dx+\int_{5/4}^{7/4}e^{-1}dx + \int_0^{1/4}e^0dx+\int_{3/4}^{5/4}e^0dx+\int_{7/4}^2e^0dx\right] = \frac{52}e$$

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