Question

$\text{Let}f\left(x\right)=max\{3,x^2,\frac{1}{x^2}\}\text{for}\frac{1}{2}\le x\le 2.$
Then the value of the integral $\underset{1/2}{\overset{2}{\int }}f\left(x\right)dx$ is

• A. $\frac{11}{3}$
• B. $\frac{13}{3}$
• C. $\frac{14}{3}$
• D. $\frac{16}{3}$

Answer 1

The correct option is C $\frac{14}{3}$


$\underset{1/2}{\overset{2}{\int }}f\left(x\right)dx=\underset{1/2}{\overset{1/\sqrt{3}}{\int }}\frac{1}{x^2}dx+\underset{1/\sqrt{3}}{\overset{\sqrt{3}}{\int }}3dx+\underset{\sqrt{3}}{\overset{2}{\int }}{x}^{2}dx$
$=[-\frac{1}{x}{]}_{1/2}^{1/\sqrt{3}}+[3x{]}_{1/\sqrt{3}}^{\sqrt{3}}+[\frac{x^3}{3}{]}_{\sqrt{3}}^2=\frac{14}{3}$