Question

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

• A. $\frac{52(1- e )}{ e }$
• B. $\frac{52}{ e }$
• C. $\frac{52(2+ e )}{ e }$
• D. $\frac{104}{ e }$

Answer 1

The Correct Option is B

To evaluate the integral \(\int\limits_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) dx\), we need to consider the behavior of the greatest integer function, or floor function, denoted as \([t]\).

First, analyze the expression \([\sin(\pi x)]\) within the integral:

• The sine function, \(\sin(\pi x)\), has a range of \([-1, 1]\).
• Therefore, the greatest integer less than or equal to \(\sin(\pi x)\) is 0 for \(0 \leq \sin(\pi x) < 1\) and -1 for \(-1 \leq \sin(\pi x) < 0\).
• As \(\sin(\pi x)\) oscillates between -1 and 1 over each interval of length 2, it remains 0 for half of the interval and -1 for the other half.

Next, analyze the expression \(e^{[\cos(2\pi x)]}\):

• Cosine function, \(\cos(2\pi x)\), also oscillates between -1 and 1.
• This implies that \([\cos(2\pi x)]\) is 0 for \(0 \leq \cos(2\pi x) \leq 1\) and -1 for \(-1 \leq \cos(2\pi x) < 0\).
• The value of \(e^{[\cos(2\pi x)]}\) is, therefore: 1 for half of the cosine cycle and \(1/e\) for the other half.

Evaluate the integral by considering these two functions together:

• In one complete cycle \(x \in [0, 2]\), the value of \([\sin(\pi x)] + e^{[\cos(2\pi x)]}\) adds up to 0 for \(x \in [0, 1]\), and to \(-1 + 1/e\) for \(x \in [1, 2]\).
• The area under the curve from 0 to 2 is thus: \(0 \times 1 + \left(-1 + \frac{1}{e}\right)\times 1 = -1 + \frac{1}{e}\).
• This interval repeats from -3 to 101, and the total length is 104, making 52 complete cycles (as each cycle is of length 2).

Calculate the total integral over multiple cycles:

• The integral over a single cycle is \(-1 + \frac{1}{e}\).
• Over 52 cycles, this sum is \(52 \times \left(-1 + \frac{1}{e}\right) = 52 \times \left(\frac{1 - e}{e}\right)\).

Thus, the value of the integral is:

• \(-\frac{52(e-1)}{e} = \frac{52}{e}\).

The correct answer is \(\frac{52}{e}\), which corresponds to option b.