If _-a^a(|x|+|x-2|),dx=22 (a2) and [x] denotes the greatest integer x, then _a^-a(x+[x]),…

Question

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

Answer: 3

Accepted Answer

For $a>2$: $\displaystyle\int_{-a}^a|x|\,dx=a^2$, and $\displaystyle\int_{-a}^a|x-2|\,dx = 4+a^2$ (splitting at $x=2$). So

$$a^2+(4+a^2)=22 \implies 2a^2=18 \implies a=3$$

Now $\displaystyle\int_{-3}^3 x\,dx=0$ (odd), and using the standard result $\displaystyle\int_{-n}^n[x]\,dx=-n$ for integer $n$: $\displaystyle\int_{-3}^3[x]\,dx=-3$. So

$$\int_{-3}^3\big(x+[x]\big)\,dx = -3 \implies \int_3^{-3}\big(x+[x]\big)\,dx = 3$$

Solutions · 1