Answer
$$\frac{{{x^6}}}{2} - \frac{{5{x^{10}}}}{2} + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {3{x^5} - 5{x^9}} \right)} dx \cr
& {\text{by the power rule for indefinite integrals}} \cr
& {\text{ = }}\frac{{3{x^{5 + 1}}}}{{5 + 1}} - \frac{{5{x^{9 + 1}}}}{{9 + 1}} + C \cr
& {\text{ = }}\frac{{3{x^6}}}{6} - \frac{{5{x^{10}}}}{{10}} + C \cr
& {\text{ = }}\frac{{{x^6}}}{2} - \frac{{5{x^{10}}}}{2} + C \cr
& \cr
& \cr
& {\text{check the antiderivative by differentiation}} \cr
& {\text{ = }}\frac{d}{{dx}}\left( {\frac{{{x^6}}}{2} - \frac{{{x^{10}}}}{2} + C} \right) \cr
& {\text{ = }}\frac{{6{x^5}}}{2} - \frac{{10{x^9}}}{2} + 0 \cr
& {\text{simplify}} \cr
& {\text{ = }}3{x^5} - 5{x^9} \cr} $$
