NIMCET 2022 — Mathematics PYQ

Evaluation of the Integral 
The integral to be evaluated is ${\int }_a^b(|x-a|+|x-b|)dx$. 
1. The absolute value function $|x-a|$ is considered. Since $x$ is in the interval $[a,b]$, it is known that $x\ge a$, which implies $x-a\ge 0$. Therefore, $|x-a|$ can be replaced by $x-a$. 
2. The absolute value function $|x-b|$ is considered. Since $x$ is in the interval $[a,b]$, it is known that $x\le b$, which implies $x-b\le 0$. Therefore, $|x-b|$ can be replaced by $-(x-b)$, which simplifies to $b-x$. 
3. The integrand is simplified by substituting these expressions: $(x-a)+(b-x)=b-a$. 
4. The integral is then rewritten as ${\int }_a^b(b-a)dx$. 
5. The constant $(b-a)$ is taken out of the integral: $(b-a){\int }_a^{b}1dx$. 
6. The integral of $1$ with respect to $x$ is $x$. The definite integral is evaluated from $a$ to $b$: $(b-a)[x{]}_a^b$. 
7. The limits of integration are applied: $(b-a)(b-a)$. 
8. The final expression is simplified to $(b-a{)}^2$. 
Final Answer 
The correct option is (d) $(b-a{)}^2$.