Answer
${x^3} +2{x^2} + x + C$
Work Step by Step
$$\eqalign{
& \int {\left( {3{x^2} + 4x + 1} \right)} dx \cr
& {\text{Use the sum rule for integration}} \cr
& = \int {3{x^2}} dx + \int {4xdx} + \int {dx} \cr
& {\text{Pull out the constants}} \cr
& = 3\int {{x^2}} dx + 4\int {xdx} + \int {dx} \cr
& {\text{Use the power rule for integration }}\int {{x^n}dx} = \frac{{{x^{n + 1}}}}{{n + 1}} + C \cr
& = 3\left( {\frac{{{x^{2 + 1}}}}{{2 + 1}}} \right) + 4\left( {\frac{{{x^{1 + 1}}}}{{1 + 1}}} \right) + \frac{{{x^{0 + 1}}}}{{0 + 1}} + C \cr
& {\text{Simplify}} \cr
& = 3\left( {\frac{{{x^3}}}{3}} \right) + 4\left( {\frac{{{x^2}}}{2}} \right) + x + C \cr
& = {x^3} + 2{x^2} + x + C \cr} $$
