Answer
$$ - \frac{{{x^{ - 2}}}}{2} - \frac{{12{x^{5/4}}}}{5} + \frac{8}{3}{x^3} + C$$
Work Step by Step
$$\eqalign{
& \int {\left[ {{x^{ - 3}} - 3{x^{1/4}} + 8{x^2}} \right]dx} \cr
& {\text{By the theorem 4}}{\text{.2}}{\text{.3 }}\left( b \right){\text{ and }}\left( c \right) \cr
& = \int {{x^{ - 3}}dx} - \int {3{x^{1/4}}} dx + \int {8{x^2}} dx \cr
& {\text{By the theorem 4}}{\text{.2}}{\text{.3 }}\left( a \right) \cr
& = \int {{x^{ - 3}}dx} - 3\int {{x^{1/4}}} dx + 8\int {{x^2}} dx \cr
& {\text{applying the power rule }}\int {{x^n}} dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C \cr
& = \frac{{{x^{ - 2}}}}{{ - 2}} - 3\left( {\frac{{{x^{5/4}}}}{{5/4}}} \right) + 8\left( {\frac{{{x^3}}}{3}} \right) + C \cr
& {\text{Simplifying}} \cr
& = - \frac{{{x^{ - 2}}}}{2} - \frac{{12{x^{5/4}}}}{5} + \frac{8}{3}{x^3} + C \cr} $$
