Key Concepts and Formulas

1. Linearity of Definite Integrals: For integrable functions $f(x)$ and $g(x)$, $\int_a^b (f(x) + g(x)) dx = \int_a^b f(x) dx + \int_a^b g(x) dx$. This allows us to break down the integral into simpler parts.
2. Greatest Integer Function (GIF): $[t]$ is the greatest integer less than or equal to $t$. The GIF is a step function, and to integrate it, we must split the interval of integration at points where the argument of the GIF crosses an integer.
3. Modulus Function: $|x|$ is defined as $x$ for $x \ge 0$ and $-x$ for $x < 0$. To integrate $|x|$, we split the interval of integration at $x=0$.

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Step-by-Step Solution

We are asked to find the value of $8 \cdot \int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx}$. Let $I = \int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx}$.

Step 1: Decompose the Integral using Linearity

Using the linearity property of definite integrals, we can split the integral $I$ into two separate integrals: $I = \int\limits_{ - {1 \over 2}}^1 {[2x]\,dx} + \int\limits_{ - {1 \over 2}}^1 {|x|\,} dx$ Let $I_1 = \int\limits_{ - {1 \over 2}}^1 {[2x]\,dx}$ and $I_2 = \int\limits_{ - {1 \over 2}}^1 {|x|\,} dx$. Then $I = I_1 + I_2$.

Step 2: Evaluate the Modulus Function Integral ($I_2$)

The modulus function $|x|$ has a change in definition at $x=0$. Since the interval of integration is $[-1/2, 1]$, which includes $0$, we must split the integral at $x=0$. For $x \in [-1/2, 0)$, $|x| = -x$. For $x \in [0, 1]$, $|x| = x$.

Thus, we can write $I_2$ as: $I_2 = \int\limits_{ - 1/2}^0 {(-x)\,dx} + \int\limits_0^1 {(x)\,dx}$

Now, we evaluate each part: $I_2 = \left[ { - \frac{x^2}{2}} \right]_{ - 1/2}^0 + \left[ {\frac{x^2}{2}} \right]_0^1$ $I_2 = \left( { - \frac{(0)^2}{2} - \left( - \frac{(-1/2)^2}{2} \right)} \right) + \left( {\frac{(1)^2}{2} - \frac{(0)^2}{2}} \right)$ $I_2 = \left( {0 - \left( - \frac{1/4}{2} \right)} \right) + \left( {\frac{1}{2} - 0} \right)$ $I_2 = \left( {0 - \left( - \frac{1}{8} \right)} \right) + \frac{1}{2}$ $I_2 = \frac{1}{8} + \frac{1}{2}$ To add these fractions, we find a common denominator: $I_2 = \frac{1}{8} + \frac{4}{8} = \frac{5}{8}$

Step 3: Evaluate the Greatest Integer Function Integral ($I_1$)

The greatest integer function $[2x]$ changes its value whenever $2x$ crosses an integer. The interval of integration for $x$ is $[-1/2, 1]$. Therefore, the interval for $2x$ is $[-1, 2]$. We need to split the integral at points where $2x$ is an integer. These points are $2x = -1, 0, 1, 2$. This corresponds to $x = -1/2, 0, 1/2, 1$. We split the integral $I_1$ over the intervals defined by these points within $[-1/2, 1]$:

• For $x \in [-1/2, 0)$: $2x \in [-1, 0)$. Thus, $[2x] = -1$.
• For $x \in [0, 1/2)$: $2x \in [0, 1)$. Thus, $[2x] = 0$.
• For $x \in [1/2, 1)$: $2x \in [1, 2)$. Thus, $[2x] = 1$.
• At $x=1$, $2x=2$, so $[2x]=2$. A single point does not affect the value of a definite integral.

So, we can write $I_1$ as: $I_1 = \int\limits_{ - 1/2}^0 {(-1)\,dx} + \int\limits_0^{1/2} {(0)\,dx} + \int\limits_{1/2}^1 {(1)\,dx}$

Now, we evaluate each part: $I_1 = [-x]_{ - 1/2}^0 + [0]_{0}^{1/2} + [x]_{1/2}^1$ $I_1 = (-(0) - (-(-1/2))) + (0 - 0) + (1 - 1/2)$ $I_1 = (0 - 1/2) + 0 + 1/2$ $I_1 = -1/2 + 1/2 = 0$

Step 4: Combine the Results

Now we find the total value of $I$ by adding $I_1$ and $I_2$: $I = I_1 + I_2 = 0 + \frac{5}{8} = \frac{5}{8}$

Step 5: Calculate the Final Value

The question asks for the value of $8 \cdot I$: $8 \cdot I = 8 \cdot \frac{5}{8} = 5$

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Common Mistakes & Tips

• Incorrectly Identifying Intervals: Ensure that you correctly identify the points where the argument of the greatest integer function and the modulus function cross integer values or zero, respectively, within the given integration interval.
• Arithmetic Errors with Signs: Pay close attention to signs when evaluating the definite integrals, especially when dealing with negative limits of integration or negative values within the functions.
• Forgetting to Split: The most critical step for these types of functions is to split the integral at the points where their definitions change. Failing to do so will lead to an incorrect result.

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Summary

The problem requires evaluating a definite integral involving the greatest integer function and the modulus function. We first use the linearity of integration to split the integral into two parts. For the modulus function, we split the integral at $x=0$. For the greatest integer function, we split the integral into sub-intervals where the value of $[2x]$ is constant. After evaluating each sub-integral and summing the results, we obtain the value of the original integral. Multiplying this by 8 gives the final answer.

The final answer is $\boxed{5}$.