Consider the following.

${\displaystyle \underset{1}{\overset{9}{}}}\frac{4x^2}{\sqrt[\phantom{2}]{x}}dx$

Use algebra to rewrite the integrand using rational exponents.

${\displaystyle \underset{1}{\overset{9}{}}}\frac{4x^2}{\sqrt[\phantom{2}]{x}}dx={\displaystyle \underset{1}{\overset{9}{}}}(4x^{\frac{1}{2}}{x}^{\frac{3}{2}})dx$

${\displaystyle \underset{1}{\overset{9}{}}}\frac{4x^2}{\sqrt[\phantom{2}]{x}}dx={\displaystyle \underset{1}{\overset{9}{}}}(4x^{-\frac{1}{2}}{x}^{\frac{3}{2}})dx$

${\displaystyle \underset{1}{\overset{9}{}}}\frac{4x^2}{\sqrt[\phantom{2}]{x}}dx={\displaystyle \underset{1}{\overset{9}{}}}(4x^{\frac{1}{2}}{x}^{-\frac{1}{2}})dx$

${\displaystyle \underset{1}{\overset{9}{}}}\frac{4x^2}{\sqrt[\phantom{2}]{x}}dx={\displaystyle \underset{1}{\overset{9}{}}}(4x^{\frac{3}{2}}{x}^{\frac{5}{2}})dx$

${\displaystyle \underset{1}{\overset{9}{}}}\frac{4x^2}{\sqrt[\phantom{2}]{x}}dx={\displaystyle \underset{1}{\overset{9}{}}}(4x^{\frac{1}{2}}{x}^{\frac{5}{2}})dx$

Evaluate the integral.