Answer
$$3{u^{ - 2}} - 4{u^2} + 1$$
Work Step by Step
$$\eqalign{
& \int {\left( {3{u^{ - 2}} - 4{u^2} + 1} \right)} du \cr
& {\text{by the power rule for indefinite integrals}} \cr
& {\text{ = }}\frac{{3{u^{ - 1}}}}{{ - 1}} - \frac{{4{u^3}}}{3} + u + C \cr
& {\text{ = }} - 3{u^{ - 1}} - \frac{{4{u^3}}}{3} + u + C \cr
& {\text{ = }} - \frac{3}{u} - \frac{{4{u^3}}}{3} + u + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{du}}\left( { - 3{u^{ - 1}} - \frac{{4{u^3}}}{3} + u + C} \right) \cr
& {\text{ = }} - 3\left( { - 1} \right){u^{ - 2}} - \frac{{4\left( 3 \right){u^2}}}{3} + 1 + 0 \cr
& {\text{simplify}} \cr
& {\text{ = }}3{u^{ - 2}} - 4{u^2} + 1 \cr} $$