NIMCET 2018 — Mathematics PYQ

Concept:
- Definite Integral: If $\int f(x)dx=g(x)+C$, then ${\int }_a^{b}f(x)dx=[g(x){]}_a^b=g(b)-g(a)$.
- If a ≤ c ≤ b, then ${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$.
- $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$.

Calculation:

Since the given function is a multi-valued function, let us separate the given definite integral into parts where the expressions of the function are different:
$${\int }_{-2}^{3}f(x)dx={\int }_{-2}^{0}f(x)dx+{\int }_0^{2}f(x)dx+{\int }_2^{3}f(x)dx$$
$$<br>={\int }_{-2}^0(x+2)dx+{\int }_0^2(2-x)dx+{\int }_2^3(x-2)dx$$
$$={\left[\frac{x^2}{2}+2x\right]}_{-2}^0+{\left[2x-\frac{x^2}{2}\right]}_0^2+{\left[\frac{x^2}{2}-2x\right]}_2^3$$
$$<br>=[0-(2-4)]+[4-2-0]+\left[\frac{9}{2}-6-(2-4)\right]$$
$$=2+2+\frac{9}{2}-4$$
$$<br>=4.5$$