Answer
$$\frac{{105}}{4}$$
Work Step by Step
$$\eqalign{
& \int_{ - 2}^3 {\left( {{x^3} - 2x + 3} \right)} dx \cr
& {\text{integrate by using }}\int_a^b {{x^n}} dx = \left( {\frac{{{x^{n + 1}}}}{{n + 1}}} \right)_a^b \cr
& = \left( {\frac{{{x^4}}}{4} - {x^2} + 3x} \right)_{ - 2}^3 \cr
& {\text{evaluating, we get:}} \cr
& = \left( {\frac{{{{\left( 3 \right)}^4}}}{4} - {{\left( 3 \right)}^2} + 3\left( 3 \right)} \right) - \left( {\frac{{{{\left( { - 2} \right)}^4}}}{4} - {{\left( { - 2} \right)}^2} + 3\left( { - 2} \right)} \right) \cr
& {\text{simplifying, we get:}} \cr
& = \frac{{81}}{4} - \left( { - 6} \right) \cr
& = \frac{{105}}{4} \cr} $$
