Question

If $f\left(x\right)=\{\begin{array}{cc}x+3,& x<3\\ 3x^2+1,& x\ge 3\end{array}$
then the value of $\underset{2}{\overset{5}{\int }}f\left(x\right)dx$ is:

• A. $\frac{211}{2}$
• B. $\frac{211}{4}$
• C. $\frac{422}{2}$
• D. $\frac{211}{6}$

Answer 1

The correct option is A $\frac{211}{2}$

$I=\underset{2}{\overset{5}{\int }}f\left(x\right)dx=\underset{2}{\overset{3}{\int }}f\left(x\right)dx+\underset{3}{\overset{5}{\int }}f\left(x\right)dx$
$=\underset{2}{\overset{3}{\int }}(x+3)dx+\underset{3}{\overset{5}{\int }}(3x^2+1)dx={[\frac{x^2}{2}+3x]}_2^3+[x^3+x{]}_3^5$
$=\frac{9-4}{2}+3(3-2)+5^3-3^3+5-3=\frac{211}{2}$