Answer
$$2{t^5} - 3\ln \left| t \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{10{t^5} - 3}}{t}} dt \cr
& = \int {\left( {\frac{{10{t^5}}}{t} - \frac{3}{t}} \right)dt} \cr
& = \int {\left( {10{t^4} - \frac{3}{t}} \right)dt} \cr
& {\text{sum rule}} \cr
& = \int {10{t^4}dt} - 3\int {\frac{1}{t}dt} \cr
& {\text{integrate}} \cr
& = 10\left( {\frac{{{t^5}}}{5}} \right) - 3\ln \left| t \right| + C \cr
& = 2{t^5} - 3\ln \left| t \right| + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{dt}}\left( {2{t^5} - 3\ln \left| t \right| + C} \right) \cr
& {\text{ = }}\frac{d}{{dt}}\left( {2{t^5}} \right) - 3\frac{d}{{dt}}\left( {\ln \left| t \right|} \right) + \frac{d}{{dt}}\left( C \right) \cr
& {\text{ = }}10t - 3\left( {\frac{1}{t}} \right) + 0 \cr
& = 10t - \frac{3}{t} \cr
& subtract \cr
& = \frac{{10{t^2} - 3}}{t} \cr} $$
