Answer
a) $2$
b) $6$
Work Step by Step
Recall: If $f(x)$ is above (or below) the axis for $a\leq x\leq b$ and $A$ is the area bounded by the curve of $f$ and the $x-$axis for $a\leq x\leq b$, then $A=\int_a^bf(x)dx$ (or $A=-\int_a^bf(x)dx$).
Using this knowledge above, we have $\int_1^3f(x)dx=A$, $\int_3^4 f(x)dx=-B$, and $\int_4^5f(x)dx=C$.
Part a)
Using the properties for Integrals,
$\int_1^5 f(x) dx=\int_1^3f(x)dx+\int_3^4 f(x)dx+\int_4^5f(x)dx=A+(-B)+C=3+(-2)+1=2$
Part b)
Using the properties for Integrals,
$\int_1^5 |f(x)| dx=\int_1^3|f(x)|dx+\int_3^4 |f(x)|dx+\int_4^5|f(x)|dx$
$=\int_1^3f(x)dx+\int_3^4-f(x)dx+\int_4^5f(x)dx$
$=\int_1^3f(x)dx+(-\int_3^4 f(x)dx)+\int_4^5f(x)dx$
$=A+B+C$
$=3+2+1$
$=6$