Answer
$$12$$
Work Step by Step
$$\eqalign{
& \int_1^2 {\left( {3{x^2} + 5} \right)} dx \cr
& {\text{integrate by using }}\int {{x^n}} dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C{\text{ and }}\int {dx} = x + C \cr
& = \left( {3\left( {\frac{{{x^{2 + 1}}}}{{2 + 1}}} \right) + 5\left( {\frac{{{x^{0 + 1}}}}{{0 + 1}}} \right)} \right)_1^2 \cr
& = \left( {3\left( {\frac{{{x^3}}}{3}} \right) + 5x} \right)_1^2 \cr
& = \left( {{x^3} + 5x} \right)_1^2 \cr
& {\text{use fundamental theorem of calculus }}\int_a^b {f\left( x \right)} dx = F\left( b \right) - F\left( a \right).\,\,\,\,\left( {{\text{see page 388}}} \right) \cr
& = \left( {{{\left( 2 \right)}^3} + 5\left( 2 \right)} \right) - \left( {{{\left( 1 \right)}^3} + 5\left( 1 \right)} \right) \cr
& {\text{simplifying}} \cr
& = \left( {8 + 10} \right) - \left( 6 \right) \cr
& = 18 - 6 \cr
& = 12 \cr} $$
