Question

If $f\left(x\right)=|x^2-5x+4|$, then value of ${\int }_0^{2}f\left(x\right)dx$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is D $3$

$f\left(x\right)=|x^2-5x+4|=|x^2-4x-x+4|=|x(x-1)-1(x-4)|=\left|(x-1)(x-4)\right|\therefore {\int }_0^{2}f\left(x\right)dx={\int }_0^1(x-1)(x-4)dx-{\int }_1^2(x-1)(x-4)dx$
$={[\frac{x^3}{3}-\frac{{5x}^2}{2}+4x]}_0^1-{[\frac{x^3}{3}-\frac{{5x}^2}{2}+4x]}_1^2$
$=\frac{1}{3}-\frac{5}{2}+4-\frac{8}{3}+\frac{20}{3}-8+\frac{x^3}{3}-\frac{5}{2}+4=3$