Animated Solution for Mathematics - Definite Integration: 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

The function $f(x)=|x|+|x-2|$ acts as a piecewise function that changes its definition based on the critical points $x=0$ and $x=2$. Given the integration limits from $-a$ to $a$ with $a>2$, we partition the integral into three distinct intervals:

1. For $x\in [-a,0]$, both terms are negative: $|x|+|x-2|=-x-(x-2)=-2x+2$. 2. For $x\in [0,2]$, the function simplifies: $|x|+|x-2|=x-(x-2)=2$. 3. For $x\in [2,a]$, both terms are positive: $|x|+|x-2|=x+x-2=2x-2$.

We set up the integral sum based on these intervals and equate it to $22$:

Evaluating these integrals step-by-step:

${\int }_{-a}^0(-2x+2)dx=[-x^2+2x{]}_{-a}^0=0-(-a^2-2a)=a^2+2a$ ${\int }_0^{2}2dx=[2x{]}_0^2=4$ * ${\int }_2^a(2x-2)dx=[x^2-2x{]}_2^a=(a^2-2a)-(4-4)=a^2-2a$

Summing these results gives: $(a^2+2a)+4+(a^2-2a)=22$

Simplifying the equation above, the linear terms cancel out:

Since the problem constraints specify $a>2$, we discard the negative root and conclude that $a=3$.

We now evaluate the integral ${\int }_{-3}^3(x+[x])dx$. By the linearity of integration, we split this into two parts:

The first term, ${\int }_{-3}^{3}xdx$, involves an odd function integrated over symmetric limits, which evaluates to $0$.

For the second term, we evaluate the greatest integer function as a sum of steps:

The final result of the integration is $-3$.