Question

If \(\int_{-a}^{a} (|x| + |x-2|) dx = 22, (a>2)\) and \([x]\) denotes the greatest integer \(\le x\), then \(\int_{a}^{-a} (x + [x]) dx\) is equal to ______

Hint:

$\int_{-n}^{n} [x] dx = -n$. This property is very useful for symmetric limits in definite integration.

Answer 1

Correct Answer: 3

Step 1: \(\int_{-a}^{a} |x| dx = a^2\).
Step 2: \(\int_{-a}^{a} |x-2| dx = \int_{-a-2}^{a-2} |u| du = \frac{(-a-2)^2 + (a-2)^2}{2} = a^2 + 4\).
Step 3: \(a^2 + a^2 + 4 = 22 \Rightarrow 2a^2 = 18 \Rightarrow a = 3\).
Step 4: Find \(I = \int_{3}^{-3} (x + [x]) dx = - \int_{-3}^{3} (x + [x]) dx\).
Step 5: \(\int_{-3}^{3} x dx = 0\). \(\int_{-3}^{3} [x] dx = -3-2-1+0+1+2 = -3\).
Step 6: \(I = - (0 + (-3)) = 3\).